1. How old is Bart, actually, 10?

2. Is Smithers really gay? I've got two instances I can think of that point to this assumption.

3. Why did "Dr. Nick" come into picture? Did the other doctor that laughed at everything die, or something?

4. Lastly, I just wanted to say, I'm gonna use this topic to post other questions that I ever think of in the future.

Thanks.

1. How old is Bart, actually, 10?

Yes, he'll forever be 10 years old.

2. Is Smithers really gay? I've got two instances I can think of that point to this assumption.

Yes.

3. Why did "Dr. Nick" come into picture? Did the other doctor that laughed at everything die, or something?

I don't know why they brought on Dr. Nick, but Dr. Hibbert (the doctor who laughs at everything) is alive and well. Dr. Monroe is the one who died. No explanation was ever given for his death.

Which episode mentioned Dr. Marvin Monroe's death?

No episode directly mentioned it, but in one episode there is a sign in the hospital that reads: 'Marvin Monroe Memorial Wing'

That, and the fact that he hasn't been seen in 13 years.

They make a lot of jokes about Monroe in the show, I'm really curious as to why they dropped his character.

They (the actors, the writers, and the producers) hated Monroe's
nasal voice. I heard this on the DVD commentary.

So this "Dr. Monroe" was strictly first season? No wonder I can't remember him..

He did have one appearance in the second season. It was the episode where Itchy and Scratchy were banned.