Starts, runs rough for 30 seconds, then it shuts off or just will not start?

Hello, before replacing any more components, if you are concerned about a fueling issue, and I have personally experienced some issues on Volvo's and the fuel systems. I would do a fuel pressure test to verify that the pressure is staying where it should at all times, meaning under a load when the vehicle is in drive, but you have your foot on the gas, which will put a small load on the engine. With these systems, if I remember correctly, you have to T into the fuel line going to the Fuel Rail where the fuel injectors are. I'll put a diagram showing this.
Another thing is if you want to try to maintain your own vehicles, you will want to buy a scan tool, even a basic one that can read engine computer diagnostic codes and show you live data from the engine while running. This is a huge help to see how the vehicle is reacting. I will post more information here for you in a minute.

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

Yes, you can get some Bluetooth type scan tools, but I recommend something like this one from Harbor Freight (4th pic). It is $99.00, but you can find cheaper ones on amazon, just as long as it says "can read live data". You don't want just a code reader. The live data is where you will be able to give us some numbers from the data and we can help you far more, knowing what the engine computer is actually doing. Sorry for this long post, but I wanted to help explain how a few things work.

With a basic scan tool that can give you live engine data; you really don't even need to take a fuel pressure reading at first. There are ways to see if the engine is running lean or rich just by live data. For example, the computer data will have data Pids (Parameter IDs) such as Long Term and Short-Term fuel trim. Or it will give readings from an Oxygen sensor, coolant temperature sensor, intake air temperature sensor, etc. Picture 3 below is an example of a few data Pids. You can see the Short Term B1S1 reading -3.9% there. That is the fuel trim from Bank 1 Sensor 1 oxygen sensor and its reading -3.9. So, the engine computer is reducing fuel from the air fuel mixture by 3.9 percent on Bank 1. This means Bank 1 is running a little rich and the computer is compensating for it.

Now from what I have seen Volvos do, they rely heavily on fueling parameters. I had a Volvo that had its rear Oxygen sensor voltage that was stuck reading full rich. The vehicle would not run at all, it would start and stall, maybe run a little bit then stall, huge lack of power. But this was all because the computer was compensating for a condition that did not exist. It was just a faulty reading. So, you can see how a simple obd2 scan tool can help in a huge way instead of changing a ton of parts. Granted the fuel filter and pump may have been on its way out. A restricted filter will ruin a fuel pump.

Having said all that. I do believe you have a fueling issue. Because of your comment about filling the tank. This has to do with the Evap system, but that is a whole other system to explain.

Another thing, you mentioned you had to charge the battery, if you drive the vehicle every day or other day, you shouldn't have to charge the battery at all. If it was reading low voltage, it is most likely starting to fail. They last about 3 years or so. But checking the battery voltage with the vehicle off and then also when running. The charging system should be 13.7 to 14.3volts. Here are some guides to help. If you have any more questions just ask. Again, Sorry for the long post.

https://www.2carpros.com/articles/how-to-check-fuel-system-pressure-and-regulator

https://www.2carpros.com/articles/car-battery-load-test

https://www.2carpros.com/articles/how-to-check-a-car-alternator

Well, since you are wanting to get into repairing your own vehicles, I'd grab a scan tool and take a look at what the engine computer is doing. Collect some data, and that will direct your next test. Some people just keep changing parts until the vehicle runs correctly. But think about how much you can save if you know exactly what part is bad. Plus, a ton of headaches. The fuel pressure test on this vehicle is very difficult, it requires special adapters, and such so save that until you're sure you need to. Let's see what the engine is actually doing first, then we can test.

Hello, I just wanted to jump in. Is this a turbo or none turbo engine? If it is turbo, I would inspect the charge air pipe. The black plastic hose that goes over the cam cover. If it popped out at the turbo side or at the charge cooler end, it would cause your symptoms of no start and give it gas to try and keep the engine running. This will also most likely set an engine light.

Another issue I saw at Volvo was the fuel pump modules getting corroded. We hardly saw them here in California, but in areas with snow and lots of rain it was an issue. The fuel pump module, if it has not been updated, is located behind the wheel well on the rear right (passenger) side of the vehicle. If it is still there inspect the connector pins and module pins for corrosion. The updated repair at the time was a new fuel pump control module and splicing in a new harness. The new location to place the module was under the rear seat.

Update us with the information and provide pictures if possible. Hope the extra information helps.

Take a look at Jorge's suggestion first, if all is okay, then we can dive into the scan tool data.

Well, the idea behind the fuel pump module is really its location, so he was referring to checking its harness wiring plug on the module. Checking inside the connector for any white or green crusty corrosion. Look deep into the female pins on the connector with a flashlight, and then the pins on the module. I can see by its location how it can get corroded easily. Honestly, I don't know why they always put fuel pump modules in the worst places. That one looks like you just need to pull part of the wheel well back to check it, you shouldn't have to remove the entire wheel well. If the connector and wiring to it looks okay, we will move on. But you can always take a look at the engine data with your scan tool.
It will be good to get used to using it, it will be a very helpful tool in the future. You can scan for diagnostic trouble codes too and see if the engine computer is setting any.
Did you get the one I put up a picture of?

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

I noticed this, on their Harbor Freight site, they have a manual you can download which will help you a lot understanding the tool. And there is also a software update for it, but I haven't checked into it yet, but I will. Looks pretty decent. Make sure the key is off before hooking it up and then key on engine off and it will start communicating with your vehicle but do still check the fuel pump module. Jorge worked for a Volvo dealership, so he will know what issues were seen the most.

Okay, that gives us some direction, since it's a circuit code we can do some testing with a multimeter, and also in live data there should be a data Pid about the mass air flow sensor, such as its grams per second (g/s), its voltage reading at idle and off idle as well as see what the other sensors are doing because of a faulty mass air flow reading. I'm not saying the sensor is bad since you mentioned already replacing it. This is a circuit code, meaning the engine computer (ECM) is seeing either a voltage or frequency discrepancy depending on how this MAF signals the ECM. The first thing to do, when you get time, is to inspect the MAF connector. Make sure there's no corrosion in the connector or loose pins. We can go over this tomorrow when you have time, I'll pull up the wiring diagrams for the MAF and see what it should be getting for a voltage supply and ground. You mentioned a solenoid you replaced; do you remember which one it was?
Also check the throttle body, make sure the throttle plates look clean around the edges where they touch the throttle body, you don't have to move the throttle plates. You should be able see if there is carbon build up by just taking the air intake tube off.

Also, what condition is the air filter in?

Yes, the throttle will need to be accessed from the bottom. Warm up the engine to soften the plastic pipe over the throttle. Will make it much easier to remove and install. If you try to install it when cold, the plastic gets very hard and very difficult to install back.

If there is oil/carbon build up the PCV system could be plugged up. Now I wouldn't think that would cause your original concern with the hard start and running rough. A defective Mass Air Flow sensor could do that. Are there any other codes that posted besides the P0101? Maybe fuel trims codes like a P0171 or P0172? Vacuum leaks will also cause that issue with P0101.

Once you get the engine running and have it idle on its own. Look at your live data. I need you to turn off all the accessories. Radio, A/C blower motor inside cabin lights, etc. I need you to give me the long- and short-term fuel trim numbers. Also give us the MAF reading in g/s Look at fuel status and let us know if it says CL or OL or it gives a weird message? We will wait for an update.

I'm still here too, we're working together on this to help you get it resolved. When you go to look at live data, write down the long term and short-term fuel trims when the engine is cold and when it gets up to temperature. Also monitor what the rear (or sensor number 2) Oxygen sensor is doing. It will be a 0.100 to 0.900mv reading. If it is stuck at either end such as 0.900mv which is almost 1volt or closer to 0.100mv write that down. Also, with the fuel trims since you are most likely just going to be looking at it in the global obd2 side it should just give you a percentage. A positive number or negative, but in case it reads a different fuel strategy, here is the service info on your vehicles Adaptive Functions (page 1).
And pages 2 to 4 are some data PIDs to look at.