I've been thinking about a lot of these same things because I'm in the process of planning a new Pokmon fanfic, so I'll answer these. Maybe it'll help both of us. ^_^
1. I think there could be events in various cities around the world where children of a certain age(likely not ten though) are given their first Pokmon. But I think you could also get Pokmon by other means, ie. a gift from a family member.
2. See above. Also note that having it once a year allows for time to breed.
3. I think gym battles would work more like league battles in the anime, ie. trainers battle each other in an elimination tournament, with the winner(s) going on to face the gym leader.
4. Gym leaders would be government workers, so they're paid with tax-payer dollars. Either that or they receive a percentage of the profits from ticket sales.
5. There would likely be designated areas around the city that could be used for battles. Think public parks, basketball courts, etc. from the real world. There would likely be restrictions on using them elsewhere however.
6. Yes. Hopefully not pay-per-view though. =/
7. Battling other trainers would likely be the preferred method.
8. I think leveling up and EXP would be social constructs, and have no basis in biology. Similar to our ages in the real world.
9. Some items would make more sense than others, but in most cases the item could be part of a charm or necklace. Leftovers though? That's a little harder to visualize.
10. I already did cover this in a story of mine, where a Kadabra evolved into an Alakazam after meditating alone in the forest for about a day. My justification was that the method of evolution was being separated from its trainer for a time, which is similar to what happens during the trading process in-game.
11. There would probably be a number of semi-conferences, with the winners of each coming together in one final event. I think it would work very similar to the real life Pokmon championships, though on a much larger scale.
12. Becoming a gym leader, perhaps? That seems like a pretty decent prize to me. Valuable items like Master Balls or evolutionary items could work too, depending on their rarity.
13. I don't see why not. I'm not sure what your reasoning for them not existing would be, so I don't know what else to say.
14. I don't care much about real world sports so I'm probably biased. but I can see Pokmon being much more popular. I've imagined that if our world had Pokmon in it, most if not all of the world's sports stadiums would instead be Pokmon gyms. Thus every US city with an NFL team would be a "Pokmon Gym City."
15. Perhaps with technology that allows for them to move within a bubble of water, which of course doesn't exist in the real world. However it would likely be developed given this alternate timeline with Pokmon in it.
16. Someone who has defeated a set number of gym leaders in that region. I don't think they'd have to defeat all of them(especially given my NFL example), so I'd go with eight, just like the games.