Yes, you can activate urgent tuning during either turn, as long as it is activated in the battle phase
If you attack a monster, and then he uses Urgent tuning and changes his side of the field, you get what is called a replay...this allows you some options.
When a replay occurs, you get to decide if you want to still attack with the same monster that declared an attack and you get to re-select a target. If you decide to attack with a different monster, then you CANNOT attack with the monster that you were attacking with because it already declared an attack, and during the replay you decided it wasn't going to attack
For making your monster stronger, it depends on WHEN you activate the card to increase your monsters ATK. If you do it during the battle step of the battle phase...he can chain using urgent tuning and decide what happens from there. If he DOES NOT use the monster that is attacking as synchro material, then the the monster that is attacking MUST continue the attack because a replay DOES NOT occur.
However, if you activate a card to increase your monsters ATK during the damage step, then he CANNOT use urgent tuning because only cards that alter a monsters attack/defense or cards that have spell speed 3 can be activated during the damge step. Urgent tuning is spell speed 2, and cannot be activated during the damge step
I am not sure about the last card...but he can use any monster in his deck as long as one is a tuner and the rest are non-tuners...this card doesn't allow him to synchro two non-tuners together to get a synchro monster. As long as the regular conditions are met for a synchro summon, then he can use urgent tunning to do the synchro summon during the battle phase.
Hope this helps