Be forewarned that the eBay Chinese replacement carbs Iif that's what you got) seem to be way out of spec and so are hard to tune based upon the methods we're used to using for the OEM carbs... jet and passage sizes appear to be different, etc., etc.
If you still have the originals and they can be cleaned up I'd suggest using them if possible.
RE the gear shifiting and reverse lever, try turning the output sprocket while attempting (or the crankshaft - CCW -through the inspection port).
I'm sure these carbs are the same. I bought them on Amazon for $70. Honestly, thus far, no complaints. I haven't taken it out riding, but buzzing around the house it feels solid through from idle to WOT. So for now, I'm not worried. I do still have the carbs that came on the bike. I will keep them in the event that it gives me trouble.
I'll see if I can look into the reverse issue and try what you say next weekend. I'm on the road for work most of the week so don't have the ability to work on anything in the evenings for the most part.