Once Kathril's ability begins resolving, no player may take other actions until it's done and you've put +1/+1 counters on Kathril.

For each applicable keyword, the appropriate counter may be put on any creature you control, including Kathril. The counters don't need to all be put on the same creature.

You check once for each keyword. If you have two creature cards with flying in your graveyard, you put just one flying counter on a creature you control, not two.

If cards in your graveyard have multiple hexproof abilities (such as hexproof, hexproof from white, and hexproof from black), a counter of each variant kind will be put onto a creature.