No you cant
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As long as you are Special Summoning before, or during the Battle Phase, then yes, that monster can attack.
You can activate any quick effect that can summon, during the battle phase. There are also many Triggered effects that would activate here too. But you cannot Normal Summon, or perform Special Summons that aren't resolvable effects, like Synchro Summons or Chaos Sorcerer.
YES!! Because you have both a main phase 1 and a main phase 2. If you do not summon a monster in main phase 1, enter your battle phase, complete your battle phase, then enter your main phase 2 and you can summon a monster with 4 stars or less unless it has a special effect or tribute the monster that already attacked. Unfortunately you will not be able to attack with the new monster until the next turn, but hey, thems the breaks.
'Immediately' is misleading, normally you summon in the main phase so you need to wait until the battle phase, it can attack in the same turn it is summoned. If a monster is summoned in the battle phase then usually it can declare an attack as the next game action.
Yes. If you have Ultimate Offering, you can use the effect during the opponent's battle phase. You can use it to do two Normal Summons, yes, you can perform a tribute summon if you wish.