If you choose the first mode of Koma's activated ability, you can choose a permanent that's already tapped as the target. The effect stopping its activated abilities will still apply.

Activated abilities contain a colon and appear in the form "[Cost]: [Effect]." Some keywords such as equip are activated abilities and will have colons in their reminder texts. Loyalty abilities of planeswalkers are also activated abilities.

You can't sacrifice Koma itself to activate its last ability, but you can sacrifice any other Serpent you control, including Koma's Coil.

A spell or ability that counters spells can still target Koma, Cosmos Serpent. When that spell or ability resolves, Koma won't be countered, but any additional effects of the countering spell or ability will still happen.

The first mode of Koma's activated ability doesn't stop static abilities from affecting the game, and it doesn't stop triggered abilities from triggering. Unlike similar effects, it does stop mana abilities from being activated. Most activated abilities that produce mana as they resolve are mana abilities.