How do you get the level 45 class gear if you use a level booster?

Says all the class quests are completed for you, but is the armor set possible to obtain without doing these quests? Or their appearance at least?

It will either be given to you or you'll have to farm any of the level 50 dungeons for the antique drops that can be exchanged for that gear.

Ok so as long as their is an option to get it I'm fine with that. Just wanted to make sure Square Enix didn't overlook this. Thank you.

You'll almost certainly be able to buy them from the Calamity Salvager as the gear shows up on it on getting the 50 Job Quest achievement. The condition for which will be fulfilled by the potion as it marks all class/job quests done.

The ilvl55-sets (the original level 45/50 quest rewards) can be bought at the calamity salavger - I would imagine that they're tied to having those quests completed, so if they're flagged "completed" in your questlog they should be open up for purchase.
For the ilvl90-set Bourne is right - you can get those, using the antique drops.
So in both cases there should be ways in place to get the gear even if its not getting placed in your inventory!

I would have to assume all quests regarding class leveling will be unlocked. And those armors are just given to you at level 45 & 50 along with the level 90 set (same look different color).

Any other additional side quests like the city dances will still be there.