It would be nice to be able to see the mobs/mobcap of different mob types (monster, animal, aquatic animal, etc.) without having to read it from the somewhat confusing list added to the F3 menu recently.

This was already added in the main development version on 1.12.2 in the beginning of this year. But the last merge from 1.12.2 to 1.13.2 and later was over a year ago, so it's not in any of the later versions yet...

Does the mobcap info also work in multiplayer?

Only if the server has a mod that sends the info. Currently this still only exists in the 1.12.2 version of MiniHUD, and in 1.12.2 Carpet mod is the only mod that currently sends that info, when requested. Sending the info will also be in the Servux mod at some point soon-ish.