Summon skeletons + WHAT?

Does support gems like add fire damage and multistrike work with skeletons?

They use a melee physical attack so any support that works with those will do the trick. Added fire isn't recommended, though, because any elemental damage they deal gets halved.

Summon skel + remove gem! ok, j/k. They do half whatever ele damage you give them so added chaos tends to be more popular. Minion instability tends to work alright with them since they die a lot. And, yes, multi strike or any other applicable gem will work with them. The best use I have found is as a distraction with spell totem. I tried to use them as DPS but they just seemed lack luster compared to classic summoner (zombies/spectres) or SRS.

last time I used them as a boss killer with Minion Instability, where I was summoning them right into boss face. So I was focusing on supports affecting explosions - minion life, minion damage, fire penetration (in +2 minion helm). But I haven't played summoner since open beta so I'm not sure how is this strategy working these days

really, it depends on what role you want your skellies to play in your army.

If you are not a dedicated summoner, then the only role I know about is distraction with spell totem or defensive trigger gems