Traveling from Lucerne to Interlaken Ost on June 7 and Interlaken Ost to (Lausanne) Montreux on June 11 via the Golden Pass with the Swiss Pass are seat reservations required? If so should I purchase them asap?
Not required - but if you want a guaranteed seat then you might want a reservation...it's up to you.
The portion that's most often reserved is if you want to be in a panorama car between Zweisimmen and Montreux: http://www.mob.ch/en/reservation-procedure.
If we do not reserve can we still use the Golden Pass for our travel? Are certain cars reserved?
Every rail line in Switzerland, including the Luzern - Interlaken - Montreux route, has at least one train per hour. the "Golden Pass train" is an extra limited stop train for tourists with panorama cars. Whichever train you travel on, you see the same mountains out of the window. If you travel by the regular hourly train, there are no reservations.
Whichever train you take (regular or "Golden Pass"), you will have to change trains en route, it is not one through train.