If a player receives a yellow card in a game, then a red card in another game (serves a 1 match suspension) then receives another yellow card in a later match. Does he have serve a suspension for the next match because of 2 yellow cards?
As suggested in the comments it depends on what tournaments are you writtend about.
Italian Serie A the player will be disqualifie after the 5th yellow cards (since last tournaments it was at the 4th). The second disqualification will be at the 10th yellow cards, then at the 14th, then at the 17th, then at the 19th, then every yellow card collected in that tournament (see the art 19 comma 9 and the box in the image below in italian language)
UEFA competitions used to disqualified the players after 3 yellow cards in 3 different matches, than after the 5th, the 7h (see art 49 and the image below) and so on until the semi-final. After the quarter-finals (art 49.04) the yellow cards are reset to do not affect the topics of the tournaments (i.e. Nedved misses EURO Champions League 2003 and Xabi Alonso misses EURO Champions League final in 2014)
2 yellow cards in different games don't account for suspension, they need to be in the same game.
However, consider this example:
In the EPL, a straight red card for a professional foul means a one match ban.
Also, 5 separate yellow cards accrued before new year also means a one match ban(extended to 10 yellows till April, after new year)
So if a player is on 4 yellows, and then gets a straight red, he will be banned for a game. (1 game for a professional foul, 3 games for violent conduct)
If he comes back, and immediately gets another yellow, he is banned again for a game.