So I have this professor this semester who I knew beforehand from a news channel my friend and I used to watch before my friend moved. This professor still is a news anchor for this news channel. I officially met this professor last semester because she was a teaching assistant in another class I had.

She surely knows who I am and has been very friendly with me since last semester when we met.
She loved the fact that I knew her as a news anchor.

I want to get her autograph during her office hours next week (last week of classes before finals here in college) because I really like her as both a professor and a news anchor and my friend would love it if I could get an autograph too.

No. I WON'T hand her a piece of garbage for her to autograph. I have glossy photo paper here at home and I'll choose my favorite photo of her from online & print it out on the glossy paper and keep it safe and not damage & bend it.

My parents also suggested that I go to this professor's hometown of Brazil with her or to visit her there, which I would love to do because I also want to go to Brazil. They also suggested that we get a meal together someday in my neighborhood because this professor has personally told me she has been to my neighborhood before. I don't really wanna do Brazil NOW, but maybe after I graduate?

So, how can I ask her for her autograph, respectfully?
Even though I'm PRETTY sure she'll accept my request and sign me one, I'm not TOTALLY sure she will accept.
I'm afraid she'll decline for her own reasons & maybe because she might think I'll share the autograph with other students of hers who DON'T know her as an anchor.
Also, how can I bring up being serious about Brazil and getting a meal with her one day, after I graduate perhaps?

I can't have my friend ask because she does not go to college with this professor and me.

Should I even ask her about these things?

Maybe wait until your class is over with her? It seems like a strange thing to ask while she is your professor.

Well, it seems like a strange thing to ask at all. I've never wanted someones autograph before so I don't see the appeal. Still, if you are going to ask I would wait until you are done in her class. It would be the lest risky time to ask in my opinion.

Well normally I'd say "geeogree has responded, end of discussion, case closed." But I guess I could put in my two cents:

If even your parents are saying you should visit her in Brazil one day, is that to say your rapport with this professor is really that strong? I think how and when you approach these questions all comes down to your personal relationship with this professor. As geeogree said, usually these types of situations are best left until after you're no longer an official student. Only then can the relationship truly progress. I'd say that's a fair answer regardless. 

But anyways, if you feel that these things could be awkward, I'd say that's a sign that your relationship may not be strong enough to go for these things while you're still a student. Asking for an autograph may seem a bit out of place, but in my opinion, who could possibly turn that down? I don't think she'd decline, but it may be weird if it's not already apparent to her that you're that big of a fan of her position. 

For the Brazil thing, you should definitely wait until you're not her student anymore. And then at that point the personal relationship will play a role, but I can't imagine any teacher not enjoying the idea of visiting with a former student. If you did build a rapport with her so far and she has the time in Brazil, she's most likely appreciate the gesture.

There are a couple of different things going on here, some are okay, some are not.

Lunch?  That's no big deal.  Maybe a little unusual but not something to stress out about.

Traveling to Brazil?  Absolutely not.  That's plain weird and she'd react negatively if you threw that out there.  That's a terrible idea.  Don't say anything about it.

The autograph with a nice photo is acceptable but I think you're thinking too much about what the teacher anticipates with your usage.

Picture it from her side.

"Hey, Eliza, I know you don't really know me but you're from Ohio?  That's awesome.  Let's travel there together to your home town."

Not cool.

Thanks guys - I got the autograph but did not mention lunch or Brazil. Everything went smoothly.