How close to closing does the TG come? If it is within a 1/2" or so, it can be forced closed usually without a problem. If it is a larger gap, you need to remove a little wood from the pad area behind the trigger. The operative word here is a "little". A little goes a long way. It should take some force to close.
As for the step on the inside of the stock. That step/cut is for the longer connector lock pin that USGI guns have. The Chinese connector lock pin is shorter, and unless the gap is filled, it will walk out and cause problems.
You can fix it one of two ways. Glue/epoxy a piece of a popsicle stick inside the cut, next to where the pin is. Or, after coating the pin and receiver with release agent, dab some epoxy into the stock and on the receive, and put them together till the epoxy sets up.