You must already control a Mountain as Arena of Glory enters the battlefield for it to enter untapped. If it enters the battlefield at the same time as a Mountain when you control no other Mountains, it will enter tapped.

If an exerted permanent is already untapped during your next untap step (most likely because an effect untapped it), exert's effect preventing it from untapping expires without having done anything.

If you gain control of another player's permanent until end of turn and exert it, and then that player regains control of it, it will untap during that player's untap step.

The mana generated with Arena of Glory's last ability can be spent on anything, not just creature spells.

If the mana generated with Arena of Glory's last ability is spent on two different creature spells, each of those spells (and resulting creatures) will gain haste until end of turn.

If mana generated with Arena of Glory's last ability is spent to pay any part of a creature spell's cost, including an alternative or additional cost, that creature spell (and the resulting creature) will gain haste until end of turn.

If mana generated with Arena of Glory's last ability is spent to cast a noncreature spell that creates tokens, those tokens won't gain haste.

If mana generated with Arena of Glory's last ability is spent on a noncreature spell that becomes a creature later in the turn, that creature won't have haste.