Team A did not break the huddle following the end of the first quarter in timely fashion. The Referee placed the ball on the floor and began the five second count. Coach from Team A tried to call a timeout and the official in front of Team A bench would not grant the timeout. I believe this is a mistake?

Also for you rule book afficianado's: Team A is down by 1 with 15 seconds to go. Dead ball and A calls time out. Everybody set ready to inbound and B calls a time out. I guess B wanted to see how defense was going to set up to be sure he got the ball inbounded. We honored it but was that correct?

Yes, this is a mistake. The ball was at the disposal of A, so the TO should have been granted/

If the TO was requested before the ball was at A's disposal, grant the TO.

As per the rule, a timeout can and should be granted in both situations.