HKMP does not synchronize anything except the players and their movement. Enemies have separate health and can be in different places. Items and abilities must be collected separately. Breakable objects must be broken separately.
Other players can still damage things in your world (including you if PVP is turned on). If an enemy in your world happens to be in the exact location where another player is hitting, it means the enemy will take damage. This is however, unlikely to happen often.
Because this game was not made to be multiplayer, there are some weird interactions that happen when your game sees other players, because they’re essentially a clone of your player model.
It’s possible to synchronize items and world objects, and provide some sort of ability to see the location of other players’ enemies. Learn which mods to install and how to set them up.
With the default behavior of the multiplayer mod, although you can’t battle the same enemy as your friend and help with progression, the mod still works well for exploring the world at the same with friends, or for PVP. If that’s good enough for you, then go for it.
If you feel like the additional synchronization provided by additional mods helps, then use that.
Although at one point a solution for entity synchronization was in the works, it does not seem like it it can be done in any reasonable timeframe as it requires a lot of manual work. The creator of the mod is exploring other options for implementing it in a more time-efficient manner.
HKMP does not touch skins from the Custom Knight mod, but there is a built-in implementation in HKMP to synchronize specifically the Knight sprites. Learn how to use skins in HKMP.