i had all the same problems as you.
try your transmission mount.
when mine was bad i'd put it in park and it would just roll down a little then come to a stop.it would not roll down at all unless i pulled down on the lever .
transmission mount can also be why the lever wont stay in 2. mine would be in 2 but always seem to end up at 3 even though it clearly was in 2nd. now 2 and 3 are very distinct positions.
as for the starting issue. check the cable on the neutral safety switch. since it obviously works, its just off adjustment, easy fix.