Minor League rosters

Yes there is, I dont know if there have been any changes but in the Pioneer league it was.
roster limit 35 active,30 dressed
eligiblity rule: no more than 17 players 21 and older,no more than 2 are 23 or older. no player on active list may have more than 3 yrs prior service.

Arizona league. 35 active, no more than 8 players 20 or over and no more than 6 players 21 and over, including 4 players not selected in current 1st yr draft. At least 10 pitchers. no more than 2 yrs prior service,excluding rookie levels out side usa

Once you are in Class A there is no age limit,but there is a limit on the amount of pro service you can have.

AA and AAA no restrictions

So what happens after your pro service years are up? Do you go into the Rule 5 Draft or have to clear waivers?

There is no restriction on age or service at the AA and AAA level.
The rule 5 draft effects players who signed at 18 after 4 seasons and players who signed 19+ after 3 seasons. Hopefully your team puts you on the 40 man roster.
Of course players can be released at any time.