Alternatively, the lifting gas can be compressed to raise the density and descend.

Why would that change anything? Neither mass nor volume (of the craft) changs.

Volume is changing somewhere. You compressed a gas bag so now normal outside air fills that space and your airship is heavier.

It's a rigid envelope, not a gas bag. The volume doesn't change, and you certainly wouldn't mix in atmospheric air.

I don't think it's infeasible that air could displace the lifting gas that was compressed, even in a rigid envelope. You could maybe build flexible bladders on the inside for this purpose. Another way to add weight using the same principle would just be to compress outside air into rigid tanks on the craft, making it heavier.

Dirigibles are rigid airframes, not rigid gas containers. They used gas bags to displace normal air from inside the frame and replace it with lifting gas.

Why would you vent it? That's what ballonets are for.