Is it because I researched better weapons too early? I know explosives destroy them, but I obviously have no control over them exploding when I shoot an alien. I have every alien and exalt weapon, but no alien grenades. I presume until I collect one, I won't be offered to research it?

Appreciate any help

The only way to collect the weapons used by an alien is to capture it for interrogation. If you have the necessary weapon needed to stun an alien, then its weapons do not explode, and some aliens (but not all of them) carry grenades. You will not be able to equip your soldiers with alien weapons until you have researched them, however.

HTH