You need to buy a separate ticket to Lauterbrunnen. You get a 25% discount with you Eurail pass. If you have a flexi pass you don't need to use a travel day for this as you get less than 100% discount on the ticket.

You only need to buy a ticket between Interlaken Ost and Lauterbrunnen. It costs 5,70 CHF (25% discount applied).

You can buy it at the train station directly.

What is the uncovered leg between Remscheid and Paris ? You'll only need seat reservations (mandatory) between Köln and Paris (Thalys, expensive 32€) or Mannheim - Paris (20€) or Strasbourg - Paris (10€). You need to buy those in advance.

The majority of railways are accepting Eurail passes, some don't (especially in touristy areas like Lauterbrunnen). There are some mandatory seat reservations on high-speed trains though : mainly in France, Italy, Spain + some cross-border services.

Can you specifiy which trains you’re looking at exactly, and on which date for the Remscheid to Paris?

It’s possible to minimise seat reservation for that, but it might take you a bit longer to get to Paris then.