Average height for antique lamp tables?

Most sit between sixty and seventy centimetres high. This puts the light source at the right level when you sit in a standard armchair. Taller tables exist but they tend to lack stability when holding heavy glass bases.

Can I put hot tea on it?

French polish reacts badly to heat and moisture. A hot mug will leave a white ring in the surface finish within seconds. Use a thick coaster.

I bought a small round table with a brass tilt-top mechanism underneath. Was this originally meant to be a lamp table or something else?

Tilt-top mechanisms usually indicate a wine table or a tea table. People folded the tops up to push them against a wall when clearing the floor. An antique lamp table rarely tips. A tipping mechanism creates unnecessary risk for a lit oil lamp. Many people use tilt-top tables for lamps today. You just need to ensure the brass catch fastens tightly.

My table has a small drawer built into the wooden frieze under the top. Would they have kept anything specific in there back then?

Small drawers held the trimming scissors and spare wicks needed to keep oil lamps burning cleanly. Some also held matches. A drawer adds weight to the top. This required the maker to use a heavier base to keep the whole structure balanced.