If you're going to do ice diving with a tether, then there are many agencies to teach it with a full set of rules and protocols.
Doing it "cave style" is a mish mash of cave rules, wreck rules and a dash of experience and lessons learned to make it as safe as possible.
We run a line just like in cave, tied off and not dragged behind us. If you haven't got the skill to do this without silting the place up, choose a different site, use a tether, or stay out of the water.
We always use a stage for ice diving. Its always a matter of when, not if a reg will free flow. Theres no issues with entanglement, it gives us an added layer of redundancy, an we're often using it for skills anyway.
We will usually disconnect our wing inflators and orally inflate, so as to avoid the potential for a wing free flow.
We choose sites that we are very familiar with, have no current and we stay shallow. Ice diving for us is the time for practicing skills and getting ready for the upcoming dive season. We'll hop in, run a line down to 30ft or so, and sit there for an hour doing drills.