Does string length affect intonation?

Yes string length does affect intonation and so does a proper set up with regard to neck, bridge, string guage, and action.

1. Do strings affect intonation?
2. Can't adjust intonation any further?
3. Does tailpiece height affect intonation?

Do strings affect intonation?

String gauge does affect intonation, but just a little bit. Having a perfectly intonated instrument is almost impossible to achieve. There are too many factors accounting for the intonation change. Action height, bridge saddle position, neck radius, nut position and height, and so on.

Can't adjust intonation any further?

You can't adjust it any further (there is a spring that holds the saddle in place that will compress up to a point). The first thing to do is to check the bow of the guitar's neck. A concave bow on your neck can throw your intonation off.

Does tailpiece height affect intonation?

It's the height of your bridge that's responsible for your action. Supposedly, lowering the tailpiece will add sustain. Raising it will gain a little tone and give a more "slinky" feel to the strings.