I always hear how hydrogen fuses into helium and when it runs out then the sun becomes a red giant as it fuses the remaining helium. My question is does it (or why doesn't it) sometimes fuse hydrogen with helium? And even if that is not a "natural process" can it or has it ever been done?

I'm going to "wave my arms" here and give a slightly educated guess.

There is a high amount of energy required to fuse hydrogen atoms (including deuterium and tritium). There would be a much larger amount of activation energy required to fuse hydrogen with helium to form lithium so it would be less common. It is probably simply that the rate of fusion of deuterium is much faster than the fusion of helium with hydrogen -- because of the activation energy.

Apparently, lithium is found in some stars and is presumably made by fusion.

Brown Dwarf Lithium Dwarfs Hydrogen Fusion Stars Temperature (http://www.economicexpert.com/a/Brown:dwarf.htm)

Some lithium was formed in the big bang.

Hydrogen - Helium fusion would creat lithium 5, which has a halflife of order 10^{-22} s. If it forms, you'll never see it.