You can do them at any order if I recall, well according to map progression at least. Except for 3 of the dungeons which need you to clear a certain amount of dungeons to unlock, 2 being at the inn and the dungeon under the castle. You'll unlock it after a certain point when tomcat tells you to go to the castle, you can get the key to the dungeon from the King after beating *alice*. If you can't read Japanese this happens sometimes after you first meet the *Alice* looking character and after? the first dungeon at the Inn if I recall. The only other one thats a bit more hidden than the rest is the one in the Mansion, to get the key to it talk to the owner of the mansion while wearing the Maid uniform, you may or may not need to do the maid part time job from the maid next to him and talk to the maid in the kitchen before you can trigger it.
Also the quests are completely optional, only ones I didn't accidentally do were the Bikini Armor quests I think.
And the scenes you get from sleeping at the inn depends on your progression, you can get all of them in 1 go once you clear the game.