Yes that’s right. The two stations in West Hampstead are a 2 minute walk apart.

I’m not sure if Eurail tells you who the operator is, but you can search the same itinerary on a UK journey planner and it will confirm.

By the way, there’s an earlier train at 0727 from West Hampstead to Kew. It won’t show up because the connection is officially too tight, but by all means take it if you reach the platform on time! You probably will

Here’s the same itinerary viewed on lner.co.uk, which is what I usually use for UK rail planning.

Trains in Britain are primarily described by time and destination.

Train numbers are not used on publicly available systems and train operator names are generally only in the footnotes on displays and departure boards.

Oh, and for platforms, not a problem. 

At St Pancras, follow signs for Thameslink. It’s down two escalators from the main concourse. There are only two platforms: A and B. You need platform B. Check the screens before you board, West Hampstead Thameslink will be the first or second stop.

Similarly, at West Hampstead Overground station, there’s only two platforms. You want platform 1 for the train to Richmond.

Thanks. Anything I should know about using my pass on the "overground"? 

Nothing special. Just add your trains to the pass.

If there are ticket barriers, it almost certainly won’t open them, but you just show it to the staff. If the barriers are open, no need.