Train from Basel to Paris, Reserve early?

There is no such thing as an "STP login". I would however create an account on the SBB website. When you do so you will have to options either "Swiss Pass" or "No Customer Number". Choose "No Customer Number". The "Swiss Pass" this site asks about has nothing to do with the "Swiss Travel Pass". It is confusing, and I do not know why they did it like that. The "Swiss Pass" is a plastic ID/customer card that is offered to residents to load commuter passes on.

So just skip that. Create an account, and use that to buy either the HFC or the STP. Note that if you buy Mürren - Paris in one transaction with the HFC it will be slightly cheaper than buying the Basel -Paris and the Mürren - Basel tickets separately. However this requires you to commit to getting the HFC. So that is first for you to decide.

TGV tickets always come with a included reservation, and apart from expensive business première tickets they are not flexible. So you must take the train you booked.

Thanks, WengenK. So if I buy the Murren to Paris ticket with the HFC, it will be on a specific TGV train from Basel?

Yes, you will get two tickets. A TGV ticket+reservation for Basel Paris, and a "connection ticket" for the Mürren - Basel part.

Will it make a difference in cost to buy this before leaving US or just buy HFC and the ticket to Paris when I arrive in GVA?

if you pay the full price now vs full price once in Switzerland there will be no difference in price.

If you buy a discounted ticket for the French train now - cheaper but restricted to a particular train, like a restricted plane ticket - vs a full price ticket in June, you will pay more for the walk up ticket, just like flying.

It is your choice which is more important to you, flexibility or price - again, just like flying.

you may well find that the train from Basel to Paris will be faster and less expensive than one from Geneva.

From Mürren the fastest route is via Basel.

Regarding when to buy the HFC: If you plan to buy it once here you can do so. You do not need to be in possession of the HFC in order to by discounted tickets. You can indicate that you have the HFC regardless of wether you have it or not. It is on the train that the conductor will check that you have it.

If I choose to purchase the train ticket Basel to Paris now, what website is the correct one?

You can buy the ticket either from the French or the Swiss Railways.

Thank you!