It is best to turn the light off at night so the bettas can rest..Mine usually sleep at night..it's cute..haha..I have one that likes to lay at the bottom though and he always freaks me out when he does that..
haha My others likes to sleep in the plants....
Also, if you leave the light on 24/7 you will most likely ..eventually run into problems with algae growth.
Lava rock isn't recommended for bettas because it has a very rough surface...You could try the nylon hose test to see if it snags..pretty sure it would...in any case...I have a few pieces of lava rock in my 10 gallon...They are medium/small rocks..All of the peices are covered fully
with java moss though...So no rough edges are exposed. I wouldn't put it in my bettas tank otherwise...Maybe if you still want to have the lava rock you could try covering it with some sort of moss.