Just addressing this question...
Is this normal?
...yes, it often is, when you manage to build up a good "fortress value". Indications that you're developing well.
I know there are ways to mitigate immigrations (mentioned already) but I just suck up everyone who arrives. (So far I've only had
one identified[1] vampire, who I burrowed out of harms way, which I don't know if is lucky or not, but obviously this is a danger. And one I was always afraid of when this addition came into the game.)
There's almost always something that needs doing (even if it's just dumping rocks away from areas you're prefer to be tidy). Take your needs as a clue as to whether to enroll more food-producers (although a legendary farmer can probably easily start off an agrarian chain of edibles enough feed the number of dwarves you currently have, possibly even overproduce) and I tend to concentrate all 'spare' effort on structure-building, which ties everyone up quite nicely. Actual experts get to mostly stick with their own field of expertise (as possible, when not requiring raw materials that I can't supply quicker by throwing dwarfpower at that issue as well) and
anybody with military skills gets enrolled in at least a
part-time military structure (and then I end up concentrating on ranged attacks from defensive positions, so most of the mêlée dwarves are unbloodied, but
really well trained). By year three (I think) my currently five-year-old fort (which I've not had time to look at, in the last few weeks) had already reached ~300, including children and babies... Many
babies!
TL;DR; Yes, it's normal. I work with it, others artificially constrain it (as discussed), some try to make sure rejects are killed off or at least more vulnerable (as also discussed). Some like it when it kicks off into one or other form of Fortress Collapse Syndrome, in keeping with the unofficial motto of the game, others see an opportunity for setting up a Perpetual Fort.
[1] By me, through the usual game anomolies, not by full in-game revelation. And so he perpetually sits, comfortable, admiring the local furniture and not wanting for sustenance, or
taking it.