Would a damaged PCV line cause hard start and running issues?

Hello, yes it can, I'll look up some service information on your vehicle, but that's a vacuum leak. And on a cold engine it requires more fuel to start and run until its warmed up, and this is causing the air fuel mixture to be lean (less fuel). I believe these engines have a MAP sensor and not a Mass air flow sensor, and a MAP sensor will be able to compensate more for vacuum leaks, but it still needs to be fixed. You may find other rubber hoses or grommets that have hardened over time and are not sealing well. Have you checked the spark plugs to see what condition they are in? And has the Check engine light come on at all? What size engine do you have? 3.8 or 3.4
Circled in orange is the MAP sensor, make sure that it is sealed into the Intake manifold very well. The wiring diagrams show a Mass air flow sensor also, do you have a sensor on the air intake boot near the air filter housing?
It would be a good idea to replace the fuel filter now also, they are pretty easy on these vehicles, it will be somewhere about halfway from the engine to the gas tank and usually on the frame rail under the vehicle in line with the fuel line feeding the engine. It will help with your starting issues. And save the fuel pump from premature failure.

https://www.2carpros.com/articles/how-to-use-an-engine-vacuum-gauge

If the check engine light is on there are diagnostic trouble codes stored in the engine computer. If you go to your local auto parts store, they will scan the computer for free and tell you what code is there. It will be something like "P0301" for example. Write down the code numbers that are stored and let us know. Then we can know exactly what is going on with the vehicle. The codes will give us diagnostic direction. Since you have a large vacuum leak like that you might have a lean code stored. The auto parts store might try to sell you the part that the code is for, but wait on that, post the code or codes here and I can tell what may be setting them. And walk you through any possible testing. This is all free, so there's nothing to lose. You don't want to damage any other components, If there are cylinder misfires happening it will eventually ruin the Catalytic Converter and they are very expensive.
Pretty much all auto parts stores will scan the vehicle for free because they want to sell you parts.
Short drives to work unfortunately are the worst for the vehicle because it doesn't get up to operating temperature fully, I know that can't be avoided, but we can at least keep it running for you. We can provide wiring diagrams and service information, recalls, etc.

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

Yes, that is the mass air flow sensor, and they usually have an Intake air temperature sensor built in. They do eventually get dirty and will under report the air flow and can cause the engine computer to not add enough fuel to the mixture. Leaning the mixture out even more.

Okay, the P0300 means there are random cylinders misfiring, not any one at specific time. But this P0128 might be the cause for the other 2 codes. This means the engine is not reaching operating temperature, and even though you have a short drive to work it should reach the set limit of 167 degrees F. The P0172 is an engine running rich code for Bank 2. It's running so rich that the engine computer's (ECM) fuel trim is below -17 percent, that's an extreme amount. These conditions will ruin the Catalytic Converter if they haven't already. So, I think you have multiple problems happening. The thermostat might be stuck open keeping the engine from warming up. And for the rich running condition, it can be a few different components. Since this vehicle is a V6, it has 2 banks. Bank 2 is the one running rich, there may be an oxygen sensor on that side of the engine that has failed and is sending a faulty signal to the ECM, if there are Catalytic Converters for each side and the bank 2 Converter has failed it can cause a rich signal, a Evap system that failed and is drawing raw fuel through the Evap Purge valve. Or leaking fuel injectors. Those are some things that can cause a rich running condition.

I put an arrow going to the Purge Valve, you can check it for any liquid fuel in it or the line coming to it. The purge valve is only to pull fuel vapors from the gas tank so they can be burned in the engine, Sometimes the Charcoal canister fails and gets filled with liquid gas instead of just vapors and they end up in the that vapor line. But the rest of the system will need to be looked at to determine the rich running condition. The cylinder misfire code will clear up after these issues are taken care of, it will save you in the long run.

You're going to have to get the low temperature issue looked at, most likely you need a new thermostat. Here are some guides below to give you an idea of how these components work:

https://www.2carpros.com/articles/how-an-engine-thermostat-works

https://www.2carpros.com/articles/replace-thermostat

https://www.2carpros.com/articles/engine-misfires-or-runs-rough

https://www.2carpros.com/articles/how-an-oxygen-sensor-works