I like your question and I think it raises a host of other questions:

How do you define the terms 'athlete' and 'sport'? Is hunting a sport? Is fighting a sport? Is racing a sport? Do there need to be spectators? What makes something a 'sport' and not just a 'game'? Would you differentiate between winning a prize and being paid just to participate? If participants bet their money (or some possession) on the outcome does that count, since they are playing for money? Does the money have to come from a third party?

How you answer these questions will go a long way in determining answers to your original questions.

Some gladiators were paid. In the Iliad, Homer mentions a chariot race in which Diomedes won a slave and a cauldron. These are just a couple of examples to show that money/prizes have been associated with sports/games/competitions for at least a little while now and you'll need some precise definitions to come up with specific answers. I'm sure others here could come up with examples from around the world over the past few thousand years depending on how the terms are defined.

But given your second question, I'll assume that you are looking more to modern sports in the industrial/post-industrial era. Maybe you aren't, but oh well, I feel like writing this post anyway. It might be fun to look at the history of professionalization in today's most popular sports. At least, I think that would be fun. Any list of the world's most popular sports will be at least somewhat arbitrary, but let's look at the sports played in what Wikipedia lists as "The Top 10 Leagues in Average Attendance". Disclaimer: I'm not suggesting that this list is definitive as "The World's Most Popular Sports", only that it included most of the sports I thought it would be fun to discuss. The list includes (in alphabetical order): American Football, Association Football, Australian Rules Football, Baseball, Canadian Football and Cricket. Because of the scope, I'll pretty much just be using Wikipedia (because it is easy) and David Goldblatt's "The Ball is Round: A Global History of Soccer" (because I read it).

With the exception of cricket, these sports became codified in the second half of the 19th century.

1 - American Football#The_first_professional_football_player)

It is hard to know when the first person was paid to play. Whenever there are rules for amateurism, payments can be made under-the-table or you can hire people for another job and have them play football for you. John Brallier was publicly acknowledged to be paid $10 to play for Latrobe Athletic Association against Jeanette Athletic Association in a game in 1895. But it was Little Billy Heffelfinger who is the first person known to have been paid to play American Football when the Allegheny Athletic Association secretly paid him $500 to play against their rivals, the Pittsburgh Athletic Club, in 1892.

In 1897, Latrobe Athletic Association paid all their players. The Arizona Cardinals of the NFL can trace their history back to 1898 when they were an amateur team named Morgan Athletic Club.

In 1902, the National Football League) was formed as the first professional American Football league. However, the National Football League that people know about is completely unrelated and was formed in 1920.

2 - Association Football

I decided to do this one last. I think I'll post something about this tomorrow. I want to read some more, but I've decided to go ahead and post what I've already written.

3 - Aussie Rules

Aussie Rules Football was codified in 1859. In 1897, eight teams established the professional Victorian Football League. And we all know the rest of that story.

4 - Baseball

In 1860, the Excelsior Club paid three players. This was done in secret. Then, in 1869, the Cincinnati Red Stockings openly paid all ten of their players a salary for 8 months. The NAPBBP was established in 1871 as the first professional baseball league. Several clubs from it went on to form the National League in 1876, which is still in existence.

5 - Canadian Football

see American Football Just kidding. Canada is not America. With that out of the way, the best I can do is to talk briefly about the Grey Cup. It was originally meant to be awarded to the top amateur Canadian Football team. It was first awarded to the University of Toronto Varsity Blues in 1909. As professionalism increased in the post-war period, the amateur unions separated themselves from the professional ones. By 1954, the Grey Cup was a fully professional thing.

6 - Cricket

While the other sports mentioned here seem to have gone from amateur to pro, cricket appears to have gone in the opposite direction. In the 17th century, cricket was tied to gambling. Rich gamblers could become patrons of professionals. Therefore, there were a lot of working/lower class professional cricketers. Okay, I don't know if there were a lot. But obviously the upper classes needed to find a way to maintain their distinction. So, those who could afford not to be paid maintained their amateur status (until 1962).

I had fun. I hope you had fun. I know that a lot of people like to concentrate on "firsts", but I hope you can see that there are some fascinating things going on with the connections between sports, colonialism, industrialization and class distinctions. Not that I was able to mention all those things. Goldblatt has some comment about how the history of sports is often seen as too historical for sports analysts and not a serious enough topic for most historians. I felt like reading his book was a fascinating way of reading the history of the late 19th through the 20th century. I'm not trying to convince you that someone was "first". I just hope you might find something here worth exploring more.