How are contracts prices determined when a new market is created? If I buy initially, am I buying directly from PreditIt? If so, at what point does the seller become another market participant instead of PredictIt? And if the market switches to other market participants selling shares, doesn't that limit the total number of shares that can exist?
Shares are always created (or destroyed) in YES/NO pairs, so there are always an equal number of YES shares and NO shares for a given contract. Because either YES or NO shares will be worth $1 when the contract closes, a YES/NO pair is always worth $1.
When you make an offer to buy a YES share for .75, this also creates a matching offer to sell a NO share for .25. If someone accepts that offer, then a new YES/NO pair is created. You pay .75 and receive the YES share. They pay .25 and receive the NO share. PredictIt receives a total of $1 and issues $1 worth of new shares.
Shares can also be retired through a similar process. When you make an offer to sell YES for .90, this also creates an offer to buy NO for .10. If that offer is matched, then you receive .90 for your YES share and the other player receives .10 for their NO share. Both shares are destroyed. PredictIt pays $1 to buy back $1 worth of shares.
Another way to think of this is that you can buy and sell pairs for $1 from PredictIt. Therefore, in order to buy YES shares, you can either buy from another player who holds YES, or you can buy a pair and then sell the NO half to someone who wants to buy NO.
I thought PI was never market participant. Instead, some people make offers, and if the first other person nabs an offer, the first market has been made.
how to suggest new markets?
When a market starts, there are no offers on either side, so it would be impossible to take someone's offer, you would have to make your own standing open offer, leave it up, and then hope someone else comes along and takes it.
None of the offers you see are ever created by PredictIt. That idea is laughable, they would never do such thing.
If you want to see what a completely empty market looks like, check this page around late morning on any weekday: https://www.predictit.org/Browse/NewArrivals