Paris to Basel (mostly rail questions)

1. The TGV train from Paris to Basel terminates in Switzerland. You should buy it to Bern. The BO pass would cover your trip from Bern to Lauterbrunnen.
2. You can use the STP to get to Lauterbrunnen. If you had a Half Price Card you would have to pay half the price of a full price train tickets from Basel to Lauterbrunnen.
3. Go with the the BO rail pass

@Jazz+Travels...I just checked the timetable for TGV-Lyria thru December 2020 and it does not go from Paris-Bern. It only goes to Basel, Geneva, Lausanne, and Zurich. So the shortest distance to Lauterbrunnen would be from Basel. Please advise. Thank you.

Basel = Swiss border. So buy a TGV ticket to Basel.
For the rest,
- STP only gives you discounts on the mountain railways in the Jungfrau, and Half fare card gives the same discounts (or perhaps better in the case of Jungfrau, but not sure about that one). If the Jungfrau area is your only Swiss destination, that rules out the STP
- I do not know the price of the BO pass so cannot advise on BO vs Half fare card, but I'm positive that the half fare card will pay for itself if you plan at least 2 excursions in the mountains, and perhaps break-even with just 1 excursion. You'll have to do the math.

1) The TGV trains from Paris run Paris Gare de Lyon --> Basel SBB --> Zürich HB
Basel SBB is the first Swiss station.

2) Yes, If you are going to Lauterbrunnen, you need to get off the TGV at Basel, and get a train from there, to Interlaken Ost and a third train to Lauterbrunnen.
You get a discount Basel to Lauterbrunnen with a Half Fare Card, but not for the section Paris to Basel.
You can buy this as one ticket (with half fare discount), or as two tickets. Shop around (SBB and SNCF), and try splitting the ticket (SNCF to Basel, SBB thereafter) or buying a through ticket (SBB all the way) to see which is cheaper.

3) You have to work it out yourself - sorry.
If you don't get a Half Fare Card, then your ticket to Lauterbrunnen needs to be undiscounted.

Two quick points:

1. Your dates seem to be set so be sure to buy your Paris-Basel TVG tickets R/T right at the 90 day mark out (earliest tickets are available) to get the cheapest fare (& it is really cheap!). You will get a reserved seat which is compulsory on TVG trains.

2. As Chris F says, the TVG to Basel leaves from Paris-Gare de Lyon station so be sure to include travel time from CDG to Gare de Lyon if you are planning to head right to Switzerland upon arrival when making your TVG reservation. Give yourself 3-4 hours from touchdown to Gare de Lyon. I suggest using a cab from CDG directly to the station. Same going back from Gare de Lyon to CDG. Bon voyage!

A little clarification on the post above: TGV booking windows recently got extended to about 120 days, so start checking 4 months ahead. If bookings aren't open yet, the SNCF website lets you schedule an alert. Trainline also works, for a tiny fee.