I've heard the "to" (in this case) referred to as "a particle". I'm not entirely convinced by this word - it seems to me no better than calling an indescribable but useful thing a "doofer" or a "thingammyjig).
As to your last question, there are some words that take only the infinitive with "to", some that take only a bare infinitive, and some that take either. There was a recent posting about this, probably in this forum, which I can't place at the moment. If the person who posted that isn't awake yet, you could make a start by looking here http://www.usingenglish.com/glossary/infinitive.html . But I'm afraid there's no alternative to simply learning/practising what goes with what.
To set your mind at rest, I'm a learner OK - just not a learner of English (though I'm a learner about English as spoken by other native speakers)