Polymerization is a spell speed 1 card. You cannot activate it in your battle phase at all.
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Not exactly. You can proceed from Main Phase 1 to the End Phase if you don't wish to have a battle phase, that is valid. You will skip the Battle Phase and Main Phase 2. But if you DO have a Battle Phase, then you must also have a Main Phase 2. A turn player can ask the opponent if it's ok to move to the end phase from the battle phase if he doesn't want to use Main Phase 2. That allows the opponent opportunity to say no, if they wished to activate something there. But the phase is not being skipped, in the same sense as the example above.
Yes, of course. Quickplay spells can be activated at most times during the battle phase and up to the end phase, the spell speed 1 ones can be activated in Main Phase 2.
For Polymerization, the Fusion Components/ Fusion Material Monsters can either be in your hand or on the field, or a combination of both.
Magical hats can only be used during your opponent's battle phase. Seeing as they can't flip their monsters during their battle phase, no. However, if your opponent does activate a spell or trap card during the battle phase, you can use Magical hats and protect your monsters. Overall, yes. You can use magical hats to protect your monsters from targeting effects as long as it's during your opponent's battle phase.
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.