Cassette size for medium derailleur

I went from 28 to 34 on a medium cage Tiagra with no change of chain (it was almost new). It's perhaps slightly shorter than ideal but still works well, and staying away from big-big helps.

I ran a seven speed system on an eight speed casette, mainly because the XT STI levers operated my cantilevers and you could not get quality seven speed casettes, or at least eight speed were easier, but I used the smaller seven cogs, my commute changed and because of one particular off road hill (Crown Prince Hill in Ash Ranges) I sometimes needed better gearing, so I shifted the derilleur to use the bigger seven cogs.  I was still using the same chain, with the length optimised for the earlier set up.  It all still worked well, until I desended a big hill on the big ring, got to a car park, dropped a couple of gears, and then came to a stop at  a road junction, went down a few more.  Terrible, I know, but I went big big and the whole drivetrain jammed, it was not easy getting the chain onto a gear so that it would work, frustrating, had to whack the rear wheel out.

If you keep the chain, DO NOT go big big.

And assuming that there is only limited wear on your current chain.

I would always work on the assumption that a new chain is required when changing cassette.

Yes and almost certainly yes, unless maybe your existing chain is staggeringly tight to the point of almost not working already.