Hi.

Im in the US and have always had manual transmission vehicles, but in the states they are becoming very rare among later model vehicles so getting people with knowledge on how to repair them is getting hard, even from the dealerships.

I have a 2017 Ford Fiesta with 5 speed mt that has 37k miles. The car has been giving me some serious issues and the Ford people, transmission shop, local
garages tell me they see so little of the manual transmissions they are unsure of what the issues are. 
I left it with a Ford dealership/service for 3 days but the car wouldnt  repeat the problems I am having with it. The only issue it
did -briefly- was not slip into reverse easily and whined on the them. They said it was a synchronizer.

The car has always had and issue with reverse. I bought it with 14k miles on it. It had a tendency to hop out of reverse or not go in easily. I baby it but it still would occur only once in awhile but wasn't too much of a problem until 3 months ago.
It started locking into gears - first then second, in neutral and in reverse. The first time was in second gear going thru an intersection. The car stuck in 2nd and the clutch went to the floor and i brake was almost non existent I was almost hit by traffic.  I turned it off and on and it cleared up. Turning the ignition off and on seems to clear the issues. I dont always notice if the the clutch or brake drops to the floor because most of the time I have had both engaged.
 

Knowing this car uses brake fluid for the clutch and brakes, I had my brother flush all the brake fluid out and replaced with fresh fluid.  seemed to help for about 2 weeks.  Again it started getting stuck in first and reverse x 2 but not second -yet.

The last two occasions the vehicle tried to continue to move, as if driving off itself, even though i had brake pressed to the floor.   Example in Reverse, it stuck and i need to stop going in reverse because traffic was coming. Once the car locks into gear it tries to keep going and then the car starts shaking violently and motor sounds like its very stressed and then i cut it off.  Again when this happened this time because i had both clutch and brake to the floor so i couldn't notice any droppage. 

Any input would really be helpful because im about ready to trade the car in - and i really dont want a new car payment.

This is a clutch fault, not a gearbox fault.

Known as clutch 'drag', it isn't fully disengaging with the pedal pressed.  This is why the car continues to pull even with the pedal pressed down.  The clutch itself is physically at the bite point.

This is generally caused by air in the hydraulic clutch line.  The fault could lie with the master cylinder behind the pedal, the slave cylinder inside the gearbox bellhousing, or a loose or damaged pipe between them.  

Sadly, there is no easy way to test the slave cylinder, the gearbox has to be removed to visually inspect it, at several hours labour cost.  Given this, I would personally recommend that the master cylinder behind the pedal is replaced first, as that's the cheapest and easiest thing to try.  If that doesn't work, then the slave should be replaced, and I would recommend having the clutch plates replaced at the same time due to the high labour cost involved in removing the gearbox.