New cassette - new chain at the same time?

Normally if I change the cassette, I'm ready for a new chain; rule of thumb is 3(?) chains per cassette when the chain checker reads 0.75.  I recently changed my cassette to a larger one which necessitated a new medium cage rear derailleur; new chain as well.  Front chain rings didn't seem worn.  Shifting was great (no noise) when I set it up on the stand, except when I went out on the road there was an annoying meshing noise in the 2 lowest gears (ie small chain ring; large cassette cogs) which I guessed was coming from the chain rings (meshing at a different angle to the previous setup?); fitted new ones and all good.  Downside of changing single components where as some people prefer to let it all wear down and change the lot (chain, chainrings, cassette).  If the cassette and chainrings have done some decent mileage and you shop around online for some components (incl chain), then it's a reasonably cheap DIY change (cheaper than the LBS) to get nice crisp shifting.

Not if you've kept an eye on the chain stretch. If it's within the limits of a stock stretch checking tool, there shouldn't be a need for a new cassette. If you've run the chain to destruction, then yes, change the cassette as well as the chain. The cassette and chain will form a 'handshake' of wear over time. Changing one without the other will force a new partnership, and during the time that's happening, you'll get a bit of accelerated  wear on the components. However, keeping an eye on chain stretch, I can easily get 3 chains per cassette, which is handy, given how much chains and cassettes cost on my group set.

cheap chain checkers are a bit naff, not all actually work, get a shimano one or measure with a ruler (not across the quick link if your chain has one)

https://road.cc/content/feature/when-should-you-replace-your-chain-219450

Not necessary unless your current chain is stretched and/or worn or if it's so tight you need longer to fit the larger cog (shouldn't be as you're only going up two teeth). I know the pro mechanic/racer mantra is generally replace chain and cassette at the same time but we don't all have team budgets or sponsored supplies - I tend to run a three chains to one cassette replacement ratio without significant deterioration in shifting performance. Chainrings should last far longer - at least the lifespan of two cassettes if not more. I have been told by a mechanic that a heuristic is new chain every 2,000kms, cassette every 5,000kms and chainrings every 10,000kms but obviously that depends on usage and type.

It's certainly good practice to fit a new chain with a new cassette, but if you're concerned about dumping what might be a perfectly usable chain, the simple question is whether the existing chain is worn. If not, go ahead and fit just the new cassette. But fitting a worn chain will risk one or both of chain slip or premature wear of the new cassette.

The only other consideration is whether the larger cassette would need an extra link. I'd guess not, unless your existing chain was already shortened to a very tight tolerance, in which case it's definitely time for a new chain.