Yes, you can. Note the exception to this (Autumn Force), in which even if you had other Powers, none of them would work.
Not automatically. If you developed Hyper-Scholarship from Mega-Int and lost your Mega-Int somehow, you'd still have your Hyper-Scholarship unless the same thing that took away your Mega-Int took it away too.
Nope. Sorry. It's really as cheap as 10 Story Points for the first Rank. You do have the spend the Willpower to develop it. I personally don't charge the additional 3, 5, or 7 Story Points for the stunt.
Hyper-Speed just lets you run really quickly, and buy lots of nifty Power Stunts, unless you're willing to spend Power Charge. In that case, you can also get extra actions by spending a point of Power Charge. One point will get your character his Rank in actions (total) per round, so if your super has Rank three Hyper-Speed, he can take three actions that round (or fewer, if desired).
Nope. Sorry. It's really as cheap as 10 Story Points for the first Rank. You do have the spend the Willpower to develop it. I personally don't charge the additional 3, 5, or 7 Story Points for the stunt.
Excellent question. No. First, you can't duplicate an exclusive Power (like Autumn Force, Domina Power, Nemesis Power, or Power Duplication itself), nor can you mimic it with Power Mimicry. It's a little more reasonable to allow Power Mimicry or Power Absorption to be duplicated, but I personally wouldn't allow it.
The answers are yes and not unless the stunt in question is from a Power that has been Perma-Burned using the stunt of the same name.
No. Just that person. Even someone with Conditional Immortality can lose his last Health Level — it just won't kill her. Everyone else in the group will die when all the Gestalt Health Levels are gone. The person with Conditional Immortality will come out of it a while later. That is, unless whatever took her last Health Level her was her weakness, and it struck her directly.

Even if someone in a Life Gestalt was killed with the thing that is the weakness of someone else with Conditional Immortality, the person with Conditional Immortality will not die, but just drop as usual.

The best optional Power for people with Life Gestalt, obviously, is Extra Health Levels. That helps everybody. Note that the Health Levels for Body Armor are not shared.
Nope. You never have to spend Power Charge if you don't want to. You might lose a point of it when you botch a roll to use an Active power, though. Some specialized uses of your Power (through Stunts, or if you want to get an extra action from Hyper-Speed, etc.) may require such an expenditure.
Excellent question. There are a number of ways to do this. The standard White Wolf method forces the person with the _worst_ initiative roll to declare his action first, and people with better initiative scores can interrupt. So, the person with the highest initiative actually declares last. Does this make sense? No, not really, but it maintains a reward for actually rolling well on your Wits + Alertness.

You could also just shrug and say, "High initiative goes first", which makes the most sense, but if the Mastermind isn't strict, can actually penalize people who roll well, as you describe. You can get around this by having players write down what they're going to do, rather than announcing it. This is a more realistic approach anyway.

You can use a variant of the system in World of Darkness: Combat, which gives little bonuses and penalties to the initiative order for certain actions. For instance, dodging might receive a + 4 bonus, so you can dodge the attack of someone who is no more than four successes ahead of you in the initiative order. I like this system a lot, but it's more bookkeeping for you.

Alternately, you can combine the first two systems. Allow the player with the highest initiative the option of going first, or saving his action until farther down in the initiative order, interrupting as he wishes.

Note that the stunt: Last Call, which is a feature of several Powers, plays havoc with the initiative system (see below).
If you're a Mastermind, then playtest it. Remember, there may be some Powers that you would only allow NPCs to have (like Total Recall, for instance), but if you find that the Power would be too unbalancing for a player to have, then revise it. If you're a player, then submit it to your Mastermind, or even to me. I'd be happy to take a look at it, and it might even become canon.
Most of the time, you can't. You can improve the Ranks of the ones you have. Sometimes, though, a Power Stunt will give you the option of buying a new Power with Story Points. There is also the Perk: Hidden Power. The Mastermind can also allow you to do so if he likes, but I caution that the character can get quite out of hand if it is easy to acquire new Powers.
No. It means that you'll have to think of some. Your Mastermind will have to approve them, of course.
No. You can usually keep your original roll. In the case of a high concentration Power, you don't have to re-roll unless you are distracted or you split your dice pool. You can't use another high or low concentration Power while maintaining a high concentration effect. For a low concentration Power, you can keep up your concentration, no problem — but you lose one die from all dice pools that do not relate to a Power (such as a Dex + Dodge roll) and your Rank in the maintained Power from the dice pools of other low concentration Powers you wish to use. Most no-concentration Powers don't have maintenance effects, but if you come across one, treat it as a low concentration Power for these purposes. The exception to this rule is the Power: Split Mind.
Not without the Power: Split Mind. In fact, you can't use a high concentration Power in conjunction even with a low concentration Power except as a Power Stunt, or unless you have either Split Mind or Synthetic Merger.
Yes, but it's bloody hard. I recommend the expenditure of a point of Willpower every time it is attempted, and an increase in difficulty for both actions. Benevolent Masterminds may allow such multi-tasking to be developed as a Power Stunt: one such Stunt for each combination that the super attempts Bastard Masterminds may require the expenditure of a point of Power Charge. This proviso also includes a few active Powers that require no activation concentration, such as Hyper-Evasion.
Yep. It works the same way as splitting your dice pool normally, unless you're talking about a maintenance effect. In that case, see above.
The super with the highest Rank, or (failing that) the super with the highest Perception, Intelligence, Wits, or Alertness, in that order. If they're identical down to the beam, the Mastermind picks the player with the nicest hair. No, I'm not kidding. Note also that X-Factoring trumps them all; a character with X-Factoring always declares last.