Battery Light is On?

It sounds like you have a alternator that is going out which is common to be intermittent. Here is a guide to help you see what you are in for when doing the job.

https://www.2carpros.com/articles/how-to-replace-an-alternator

You will need to remove the serpentine belt.

https://www.2carpros.com/articles/replace-serpentine-belt

Check out the diagrams (Below)

Please let us know if you need anything else to get the problem fixed.

Cheers

Unplug the connector on the back of the generator. If the light goes off, the voltage regulator is not turning off. Replace either the regulator or the entire generator. On the unlikely chance the light stays on when the regulator is unplugged, the green / red wire is grounded someplace. A dead short should cause the light to be bright so you would not be looking for a dead short. You'd be looking for the wire melted to another wire. Even that isn't likely.
Be sure to check all connectors and grounds for good contact

caradiodoc

Attached below is the wiring diagram for your charging system

Does the engine still run? Your engine shaking tells me something internal might have broken, or a motor mount broke. Have the computer scanned for codes in the memory. Use a voltmeter to check the voltage in the battery. Have the battery load tested to see if it holds a charge. The oil light being on normally means that the sensor is not detecting oil pressure. Make sure there is oil in it. Remove the valve cover and crank the engine over. You should see oil flowing on top of the cylinder head. No oil flow means a faulty oil pump.

Your alternator is most likely starting to fail. Your vehicle runs using the alternator, only uses the battery to start or uses the battery as back up if the alternator does fail during driving. Most part stores will test your charging system free of charge. I suggest having your battery and alternator tested.

Hi anosal, Welcome to 2carpros and TY for the donation

Check and test fuse no. 10 10amp and fuse no. 61 7.5amp if both fuses are okay -have the computer controlled field generation circuit checked and tested.

See below:

If the check enginne light is on, go to another mechanic and have them use an On-Board Diagnostid, (OBD) reader which will tell the mechanic the error codes that are triggering the check engine light.
As far as the battery light, the diagnostic might bring an error code that explains that as well.
As far as the head gasket, I would need to know why he thinks it needs it. Usually things like poor performance, smoke in the exhaust and milky engine oil are signs of a leaking head gasket.

Getting the error codes from an OBD will gi ve you more insight into the problems that the car has.

This is an intermittent problem, so any testing has to be done while the problem is occurring, otherwise everything is going to test okay.

This type of problem is usually caused by a failing generator, and this type of failure is most commonly attributed to worn internal brushes that are making intermittent contact. This can be expected at the mileage you listed.

The warning light can be misleading. It's not related to the condition of the battery. It means the charging system is not keeping the battery charged while you're driving, and that involves the generator and its built-in voltage regulator. In your application, the voltage regulator turns the "Battery" warning light on during a no-charge condition, like on all other cars, but also for an under-charge or an over-charge condition. Voltage regulators can fail in multiple ways. One way results in system voltage going too high. That might explain the bulbs flickering and burning out.

Corroded connector terminals can cause all the same intermittent symptoms as a failing generator. Check for corrosion or spread terminals in the connector at the rear / side of the generator. If you follow that wire harness, there is usually another two-wire connector about 10 to 15 inches from the generator. Check those terminals too.

While you're under the hood, follow the smaller battery positive wire to the under-hood fuse box and be sure that connection is clean and tight. That is a real common source of trouble on all car brands. When using metal tools on that connection, don't let them make contact with any metal part of the car at the same time. Doing so will result in huge sparks. Follow the smaller negative battery wire to where it bolts to the body sheet metal and be sure that connection is tight and not rusty.

You already did the correct first step, but at the wrong time. The defect only shows up once in a while. If any testing is done when the "Battery" light is not on, everything is working so there will be no defect to be found.

Have the charging system tested again, but your mechanic should understand the problem doesn't act up all the time, so the system has to be tested while the warning light is on.

All of the symptoms you described can be caused by a failing generator, but a poor connection in an electrical plug can cause the same thing. Once your mechanic confirms the problem is occurring, it should only take a few voltage tests to narrow this down to a defective generator, (alternator), or a wiring problem.

Classic bad ground issue on the Focus. While the engine is running, feel the negative cable near the battery while the headlamps are on bright, A/C on, rear window defrost on, radio on, and any other electrical load is on. More than likely it will be too hot to hold. Add extra 4 gauge wire between the body to engine, body to negative battery.

Wouldn't that also show up as slow cranking?