In general I'm a fan of the 32 oz deli containers because they've got a really wide opening so it's easier to feed without spilling flour everywhere
My kitchen has been a lowkey mess as a result of a small mouthed mason jar.
With a liquid state, try feeding it in a ratio of 1 (starter) to 2 (water) to 2 (flour) for a bit until it thickens up again.
Going to try this with my main starter jar. I transferred to 2 jars for exactly your point in #2.
Greatly appreciate the feedback in the thread. :-)