What happens when your pet reaches level 60?
Do you still gain Proficiency Points?
Can someone explain this to me?

ur pet can't gain proficiency points unless it levels up. (1 point every 5 levels.)

60 is currently the cap. your pet will not earn any more anything after lv. 60 except for experience. (Which won't lv it up either.)

When the lv. cap is raised, all experience will be reset and then you can lv. ur pet up to the new lv cap and earn more proficiency points. (remember, it's based on lv's not experience)

Experience is reset otherwise some people's pets would automatically raise them to the lv. Cap, giving them an unfair advantage.

Overall experience will still be recorded however, for the leader boards.

When the experience is reset, your pet will not lose levels. It will only reset it to 0 out of whatever is needed to reach the next level.

Thank you so much for explaining that to me, and I cant wait for the level cap to be raised :)

Does that mean there are no advantages whatsoever to filling your experience bar beyond the max once your pet reaches 60? On some other games I've played, after the max level is raised the accumulated experience boosts them to the level they would have been.

No, there are no advantages whatsoever.
(Other than keeping count of experience gained for the sake of leaderboard.)

The only reason why levels are capped is because the game is not finished; if Sylestia were updating the zones more often (and thus raising the level caps), then this wouldn't be an issue. However, the zone being released is often just a 5 or so level difference (OR a big number of zones can be released for a 20 level difference), so it'd just be fairer for everyone to start off leveling up one.

I kinda see it as stagnated growth all across the board, for the pets and the quests.