Especially when you don't have access yet?

In what sense? In that if you don't do quests one day, you can do them the next day on top of that days quests? If that's what you mean, then no.

You get random event quests right? Well I don't have access yet so I've had like 4 RE. Am I going to have 4 quests when I get access or just the most recent one or none at all?

RE quests dont accumulate, they are quests for making a bumbus.

Thank you for the help. I also need a few more posts to get a to a hundred. . . so yeah.

The Bumbus quest doesn't go away, so you are not in any hurry to get into the Blue Building either.

There is another RE that will give you (temporarily) an additional Wizard Quest, and this one doesn't accumulate either. The Ice King RE and teh Tempest REs ARE timed, they will sit there on that particular page as long as you do not refresh that particular page or get a duplicate RE for 30 minutes. I have found it useful to open these in a new Window, so I can continue to do whatever I was normally doing, and then I can come back and do them/see the results later.