Do most of Infantry units allow soldiers to use their own helmets and plate carriers?

In my experience this is typically not the case for helmets. You are welcome to purchase after market suspension systems for your helmet, but many units will have policies about using issued/authorized helmets.
Plate carriers tend to have more wiggle room within unit policies and SOPs. Often if a unit authorizes plate carriers they will authorize some off the shelf use, sometimes providing a list, as long as they accept issued plates. Other units may innsist that you use either army issue plate carriers or even IOTVs.
Of course YMMV and experiences are as varied as the units we serve in.

In general plate carriers are for a specific subset of any unit that has a specific need. Tank crew , IFV crew, the infantry teams climbing mountain tops in Afghanistan might all be authorized by unit leadership to wear a plate carrier and not full IOTV.
And all of those are going to be issued a plate carrier, that is what they will use. Now if leadership has authorized the use of a plate carrier and not full IOTV... Does anyone care if they are using the ACU pattern one issued, or a Multi cam version that holds the same issue plates in the same manner, covering the same area....nope not really.

Ive never see any SM, regular army, Guard, AF or Army reserve that was wearing a personal helmet over a issued one.
Has it happened? sure it has, right before they got body slammed for being stupid

No they do not. While most units allow non issued pouches and trucks, the body armor and helmet are vary rarely allowed to be purchased off the market.
There are several reasons for this-
Quality gear costs money, Joe should be spending $1400 just because he wants the wear the new badass ops core helmet. The one the Army gives us does its job just fine. And they are getting better all the time.
Knock off or improper gear is all over the place. Google Recondo vest and over half the results will be for airsoft or paintball use. There are too many companies selling gear for the Army to test and verify the quality and stopping power of body armor and the matching plates.
This quality control being up the last reason, investigations. If a casualty is taken, whether WIA or KIA, there will he an investigation. If the SM was not wearing Army approved helmet and body armor they will look to the commander for his risk mitigation plan. Most commanders do not want to accept this additional risk of non-Army approved plates and helmet.
Even units that once allowed non issued pouches can change depending on a new commander or cam. My last unit we could use whatever pouches we wanted that enabled us to get the job done. New BDE CSM and we were only allowed the Army issue garbage.
My advice is check with your unit, then omly purchase pouches that will help you do your job. At your level just start with an admin pouch for your status cards and a radio pouch for your radio.

You are a Corporal. That makes you a troop leader. You cannot accurately judge what condition your troops are in, and what you can ask of them, if you are not in the same gear. I ran into this problem in the 80's when high speed, low drag military looking boots were all the rage. They were about $200 a piece. Issue boots were less than $20. When I had NCO's and Lt's giving crap to junior Marines (who could not afford $200 boots), about taking care of their feet, I had enough. If it isn't issue, it doesn't belong in a leader's kit.

NO, NO, NO. in the field training maybe, but in combat- are you crazy?. 1. Does the equipment meet ans is sanctioned by DOD. 2. If your personal equipment fails, both you and your NOK could be penalized.

Army commanders should not allow SM to wear aftermarket protective gear. This is because it has not gone through army testing. and as mentioned by MSG (Join to see) the individual my not get compensation should something happen to them while wearing aftermarket gear.

From my days as a medic in the infantry I had a little wiggle room because my mission was different but I was dress right to the other medics. Standardization is what makes a difference when seconds count. For example, what is in your IFAK and where it is makes a difference when your battle buddy is trying to find a tourniquet for you in the dark with you flopping on the ground after getting shot. Even if you serve in JSOC the cool guy gear is free, if you want to look cool join the Rangers.

No

Nope, why would you spent extra money when everything is issued to you? Even as OC-Ts at NTC we were not allowed to wear commercial ACHs (Policy just been added a couple of years ago). For regular Infantry Units, I have not seen any Soldier allowed to use their own.