Question

I have a rook, my opponent has a rook and a bishop. If I don't
make any stupid mistakes, can he still checkmate me?

I think with perfect play from both opponents it's a draw. Still, there
are a lot of positions which are winning for RB-side. I want to try to
reach such a position at least for some more moves.
I don't think the creators of this site have build-in the 50-moves rule
(if within 50 moves there's no capture or pawn-move the game is
declared a draw) but I will give a draw if that turns out to be the case
(or earlier if I don't make any progress).

With respect,

Hotpawn

On a similar note, is the draw by three moves rule in effect here?

-legionnaire

Draw by repetition, draw after 50 moves without taking anything, draw
with two kings left, draw with two kings + their Knight. When those
situation happen you have to offer the draw to take it effective. The
only automatic draw is the stalemate. (well, the last time that
happened to me I had to offer the draw, but it might be resolve now...)

Yan

i did this to someone. rhp does not do it so u have to tell ur
opponent and get the draw that way

David

I think you should both be placed in a steel cage and forced to fight it
out to the death --- with a wet noodle.

Rein

I hope you're not knocking the ancient art of wet noodle fighting -
deadly in the right hands...

Rook and bishop versus a rook is a draw with best play. It's not too
easy to defend however, and a high percentage of games are won
(something like 70-75😵. Try not to let your king be driven to an
edge. Good luck.

is this discussion about an ongoing, rated game? If so, please be
careful then. Gil.

It is! See my answer