We were given a tummy time mat as a gift - I has a little pillow to wedge under the arms to help keep DD from lying flat on her face! I think a rolled up towel is supposed to do the same thing, but I could never get it right.
DD is only happy to do it for a few minutes at a time... And as much as I hate it, if the TV is on she will do whatever she can to see it (mat now faces the back of the room!). I get down on the floor and play with her and hold her fave toy up for her to look at.

How cute is tummy-to-tummy tummy time??! I love the little head peeking around... so cute!

As for getting weighed - I would think as long she is growing it wouldn't really matter. Although it is always the first question everyone asks.