My wife and I have contrasting styles. She sound just like you. She's very detail oriented. She'll make lists, and ask lots of questions. And, honestly, it drives lots of people crazy. They misunderstand her "pickiness" and never-ending questions as lack of trust, or a continuing cross-examination. But that's just the way she reaches her comfort level. I tend to be a bit more laid back.
On the other hand, if you call you do deserve a return call or text message. That's just a minimum level of responsiveness. No, you're certainly not being too demanding or unreasonable to expect a prompt response to a call, e-mail, or text.
You ask "If an agent is always too busy is that a good thing?" Depends on what "too busy" means. For some people, "too busy" might mean that she's never available when you call, even though she responds within 10 minutes or so. Other people might define it as not responding at all, or taking a day or two to get back to you. There's no one right answer, though I'd say that if you get a response in 5-10 minutes, that's very responsive. A couple of days is not responsive. I'm guessing--from your question--that you'd really like to be able to get right through to her. Recognize that that's not always going to happen.
As for liking to be very involved, that's fine. Again, though, some agents may misunderstand that or even--sometimes--not be comfortable with that. And partly that depends on how involved you want to be. But many agents (most, I'd guess) welcome client involvement. That's especially true when you're buying a home. For an agent to do a good job, he/she really has to understand you and your interests as much as possible. My only caution is to do it in a neutral manner, so your actions aren't misinterpreted as meddling or as reflecting a distrust of your agent's abilities.
How long does it take a person to find a home? It depends, ummm, on how picky they are. As well as, of course, the number of homes available in a market that meet your needs. Some buyers find a home and make an offer on it the first day they're looking. For others, it might take 6-8 months of steady looking. Just a wild guess (and I'm sure there are plenty of studies on this), something in the range of a month might be reasonable. Understand that when you first go out looking, it's an educational process for both you and your agent. You're seeing what's out there in specific price ranges. And your agent needs your feedback on what you liked and didn't like about each house you looked at. After 2-3 such sessions, you should have a good understanding of what's out there and your agent should have a good understanding of what you like and what you don't like.
I'm guessing, maybe, that it'll take you longer than average to find a property. That's OK. Just keep giving your agent feedback on what you like and what you don't like. Meanwhile, listen to your agent's response.
Finally, if you sense a disconnect between yourself and your agent, have a talk with him/her. And say very honestly, as you did in your question: "Look. I tend to ask lots of questions. On top of that, I'm a first time home buyer. And I like to be involved stuff like this. I know that those traits can drive some folks crazy. I just want you to know that I'm serious about buying, and all my questions and phone calls aren't because I'm questioning your ability or your professionalism. It's just the way I function best."
Hope that helps.