Whatever you think of Richard, he was not a fool. By that point, Edward V is an adult and Ralph Shaa, the source for the claim that Edward V is illegitmate, is dead. And barring enough butterflies, Richard's only legitmate son is dead.
I do think Richard would remarry, though not with the speed that OTL's planned marriage to Joanna of Portugal was considered. Unless the court turns against him or Edward V proves himself an inpet king in TTL, I don't see Richard going into revolt.
OTOH, if Edward V's brother Richard of Shrewsbury or his cousins John de la Pole or Edward of Warwick prove as ambitious as Warwick's father did in OTL, there may still be attempted revolts. (Many dismiss Warwick as lacking the mental capacity for that, but Henry VII thought he was dangerous enough to lock away at age 10 and execute at age 24.)
We don't know enough to answer that, he could be good, mediocre or spectacularly bad as king.
He was engaged to Anne of Brittany. If that doesn't change in TTL, France and England may already be at war over Brittany before Edward V becomes king.
There's also the problem of an heir. In OTL, Anne produced a lot of children by two different husbands. Most were stillborn or died within a month of their birth.