Is it possible to have a league with an odd number of teams while making sure each team plays the same number of games?

You cannot make a schedule that would have both an odd number of matches per team and an odd number of teams, as the product of these is also odd. Since a single match counts for two teams, you can already see where the problem would lie.

As for making a schedule with an odd number of teams and an even number of matches, that is quite easy. Assign each team a unique random number from 1 to the total number of teams. Make the following schedule for round 1:

1. 1 - x 
2. 2 - 17 
3. 3 - 16 
4. 4 - 15 
5. 5 - 14 
6. 6 - 13 
7. 7 - 12 
8. 8 - 11 
9. 9 - 10 

To get a schedule for the next round, move all of the teams clockwise.

1. 2 - x 
2. 3 - 1 
3. 4 - 17 
4. 5 - 16 
5. 6 - 15 
6. 7 - 14 
7. 8 - 13 
8. 9 - 12 
9. 10 - 11 

Repeat as necessary. Notice that after each round, there is one more team that hasn't played a game. After an even number of rounds, there will be an even number of teams, which can then be assigned among each other.

A complete 6 game schedule, for example would look as follows:

1. R1: R2: R3: R4: R5: R6: 
2.  
3. 1 - x 2 - x 3 - x 4 - x 5 - x 6 - x 
4. 2 - 17 3 - 1 4 - 2 5 - 3 6 - 4 7 - 5 
5. 3 - 16 4 - 17 5 - 1 6 - 2 7 - 3 8 - 4 
6. 4 - 15 5 - 16 6 - 17 7 - 1 8 - 2 9 - 3 
7. 5 - 14 6 - 15 7 - 16 8 - 17 9 - 1 10 - 2 
8. 6 - 13 7 - 14 8 - 15 9 - 16 10 - 17 11 - 1 
9. 7 - 12 8 - 13 9 - 14 10 - 15 11 - 16 12 - 17 
10. 8 - 11 9 - 12 10 - 13 11 - 14 12 - 15 13 - 16 
11. 9 - 10 10 - 11 11 - 12 12 - 13 13 - 14 14 - 15 

To solve the extra rounds for the teams that got byes, make a matrix of played games between them:

1. 1 2 3 4 5 6 
2. 1 x . x . x . 
3. 2 . x . x . x 
4. 3 x . x . x . 
5. 4 . x . x . x 
6. 5 x . x . x . 
7. 6 . x . x . x 

As you can see, odd teams haven't played the even teams at all, and vice versa, so a trivial solution for the extra round is

1. Rx 
2.  
3. 1 - 2 
4. 3 - 4 
5. 5 - 6 

And there you go, a schedule, where every of the 17 teams has played 6 games against different opponents.

At this point I have to point out that I have no idea how this algorithm would perform when the number of rounds is more than a half of the number of the teams. I simply didn't check it. I do know that if you repeat the loop for N-1 times, it does give the full round robin tournament schedule where every team has played every other team once.