Could be a module in the transmission, or a bad connector/wiring or a serious internal issue. If you are a DIY type of person and have the means, drop the pan and do a fluid/filter change.
It's really hard to tell over the internet with certainty. Did you check the fluid level properly? What's the mileage? You can start by getting the transmission serviced. Doubt adding a "small amount" of fluid would accomplish anything. You really need to take it to a trans shop or dealer for a proper diagnosis.
The best tire height would be one that does not exceed factory width/height so you won't run into any fitment issues. There are several online tire size calculators you can use to enter your current size then play around with the 20" sizes to get the one that matches closest in width and diameter.
It's a pretty important piece, I agree that taking it back and telling them it fell off your car after the repair was made and they need to find out where it goes and re-install it.