First there are trains with mandatory reservations but the majority of trains do not require seat reservations. The Eurail Pass is your ticket for all of them, you get a QR-code through the app by adding the train you'd like to take before boarding.

Seat reservations in Switzerland are optional and often not possible, they're a complete waste of money anyway. You simply board any train you'd like

Other point : the TGV to Switzerland has a very high fee -> 37€ 2nd class or 70€ 1st class. It is advised to avoid it. Instead take the same TGV as far as the border station, in this case Mulhouse (for both classes -> limited 10€, then 20€ until the train is full). There are also more TGVs to Strasbourg. From Mulhouse or Strasbourg take a reservation-free TER regional train across the border. You'll save a lot of money and it is only max 30 min longer (often it only reduces the waiting time in Basel)

You can book TGVs within France here (4€ booking fee per order, compared to the 2€ per person per train on eurail.com) : https://travel.b-europe.com/Eurail-GE/en/booking-tgv#TravelWish

It is advised to avoid eurail.com for booking seat reservations because they add a 2€ booking fee per person per train + they're sometimes more expensive than through the railway companies. General advice :

https://community.eurail.com/train-connections-reservations-47/how-to-get-reservations-105

https://www.seat61.com/interrail-and-eurail-reservations.htm

Let me know for questions

Schedule example :

- Paris Gare de Lyon 10:20 - Basel (37 min wait) - Lucerne 15:05 37€

- Paris Gare de Lyon 10:20 - Mulhouse (16 min connection) - Basel (12 min connection) - Lucerne 15:05 10-20€

This is an example. There are also a lot more options from Paris-Est via Strasbourg and it might even be quicker depending on where your hotel is in Paris :)

Let me know your travel date if you'd like specific advice