How to fruitfully utilize faculty-student orientation sessions at institutes you don't want to attend? More than once, I have had (graduate school admissions) orientation sessions where the faculty of the institute introduce their fields, their research and what they expect from potential students (Among other things). 

At places I am interested in, this is a useful exercise. However, there are often places which I don't wish to study at but have a few good professors doing some really good work with great knowledge of the field and awesome intuition.

How does one fully exploit an opportunity to interact with good professors from institutes you are not interested in attending?

This is very different from a research conference for obvious reasons.

Further,


If I am not interested in that institute but wanted the opinion of one of the professors about another institute/lab (the ones I am interested in attending), is that a taboo?
How does one stay in touch with such contacts where your only excuse to mail is to ask a question? I really want to be on email terms. I have read and understood their work but what is it that should be the content of mails written to them? It doesn't make sense to simply send them emails saying that I read your paper or I attended your talk.


(Please feel free to edit this question in any amount if necessary)
I don't think you can expect to get the opinion of a professor about another institute/lab. You can probably get some facts (like this other institute has more/less money, they are more/less active in this area), but nothing subjective. Academia is pretty small world, and people try as much as possible not to say anything negative publicly (and since everything said is positive, it can be hard to distinguish the real positive from the fake one). 
You need to have a real interest in their work. As mentioned here, it's already pretty hard to maintain collaboration between people interested, so if there is no clear interest, it will very hard to be on email terms. But, the question is: why do you want to be on email terms? 
See the previous point. You need to really interested, meaning you've read and understood their papers, and are able to ask questions that go beyond "what are the possible usage of your approach?". 


As a general remark, professors are usually already very busy dealing with their own research/teaching/students. Of course, they usually are open to new collaborations, but they might lack the time to work with a student who is not theirs. A good solution could to try to visit them, academically speaking, for a month or so, to work on a very specific topic.