Can't get lowest gear and derailleur shifts on its own
We know it worked once, but we don't know what was changed, so it's time to go back to basics.
Auto shifting
The rear derailleur has a spring that wants to move the chain to the smallest cog.
The shift lever should have enough friction to keep the derailleur spring from going where it wants but not be so tight that it won't move.
The first thing to check is that the wing nut is tight.
If the wing nut is tight and there is still auto shifting then it's time to check how the levers for correct assembly.
My guess is that you have shift levers that look something like this:

Check that all the washers are there and in the right place.
Low gear
There are at least three main things that can keep you from getting low gear.

Limit screws - you have checked them and they are fine. Just to make sure, can you move the derailleur by hand and get the chain to go into low gear? If so then we know it's not the limit screws.
Cable tension - with the chain in the smallest cog and the shift lever all the way down the cable should have no slack.
Derailleur alignment - here's a picture:

Picture from SOS Hanger