With cards such as these, what is the interaction with the battlefield once played?

Can you still claim? Does the player who still holds it go first the following round? Etc.

It's all so confusing!

TK

Ending a turn means ending a turn. No more claim etc. The fact that hyperspace jump switch the battlefied is not a claim in itself.

the exception would be retreat, since your opponent gets one final action, he could declare it as a claim action.

Chak

Claiming does three things; it gives you control of the battlefield; lets you use the battlefield's effect, and forces that player to pass all future actions.

Having control of the battlefield determines who goes first, and is also referenced for effects like Defensive Position or Krennic's ability.

If a round ends without someone claiming (via a card effect, or both players passing), the person who had control of the battlefield retains it. They do not get to use the battlefield's effect as they did not claim.

In the same way, taking control of the battlefield (e.g. via Lockdown) is not the same as claiming. Neither is 'Using the battlefield's claim ability' as per New Orders.

As mentioned above, Retreat allows the opponent one action, which could be to claim the battlefield.