IPO is "Initial Public Offering". Just so you know.
The valuations are done based on the company business model, intellectual property, products, market shares, revenues and profits, assets, and future projections. You know, the usual stuff.
Yes, it is. And very frequently done. In fact, I can't think of any company that is now publicly traded, that didn't start this way. The first investor, the one who founds the company, is the first one who invests in it after raising the capital (even if it is from his own bank account to pay the fees for filing the incorporation papers).
What is the difference between "normal" investor and "angel"? What do you refer to as "angel"? How is it abnormal to you? Any investor can play a role, depending on the stake he/she has in the company. If the stake is large enough - the role will be significant. If the stake is the majority - the investor will in fact be able major decisions regarding the company. How he bought the stocks, whether through a closed offering, initial investment or on a stock exchange - doesn't matter at all.
You may have heard of the term "angels" with regards to high-tech start up companies. These are private investors (not funds) that invest their own money in start ups at very early stages. They're called "angels" because they invest at stages at which it is very hard for entrepreneurs to raise money: there's no product, no real business, usually it is a stage of just an idea or a patent with maybe initial prototype and some preliminary business analysis. These people gamble, in a sense, and each investment is very small (relatively to their wealth) - tens of thousands of dollars, sometimes a hundred or two thousands, and they make a lot of these. Some may fail and they lose the money, but those that succeed - bring very high returns. Imagine investing 10K for 5% stake at Google 15 years ago.
Those people are as investors as anyone else, and yes, depending on their stake in the company, they can influence its decisions.