| Yes, you trill between the note written and and it's nearest upper neighbor - that may be a whole step or a half step depending. For instance, if you are playing in the key of C, and the written note is a B, then you would alternate between the B and the C.
Yes, the trill should always be fast, it doesn't really slow down with the tempo of the piece. If the trilled note is short, then your trill is fast but ends quickly enough to "show off" the written note. If the note is long the trill is just as fast, but you play it for longer, again always stopping on the written note long enough to let the written note sing.
I was taught that on a trill you start on the upper neighbor and try to fit in six notes in the trill so you always end on the written note. This is different than a mordent, where you start on the written note and play the upper neighbor only once, and then back to the written note. The effect is very different between the two techniques. | 
