Rail connection in Zermatt for Glacier Express

1) Yes - or you can do the whole plan an hour later, with departures 7:06 and 9:52. Another way to see these connections is to request a schedule from Bern to St. Moritz with "stopovers" at Zermatt and Disentis.
2) Yes, Zermatt only has one and it's small.
3) Yes, plenty. There is a shorter, 6-minute connection in Visp, but that should also go basically like clockwork.

You need not go all the way to Zermatt if that's not important to you. You can depart Bern at 8:06 and connect at Brig to a Glacier Express departing 9:18. This connection includes walking across a street and into a second station; see the station plan.

All of Switzerland's marketed "scenic express" trains are covered by Swiss Travel Passes. Very few trains are not fully covered in the country, though Wengen-Jungfraujoch is one that's only discounted (25%). A direct train from Bern to Interlaken is your entry point to several trains and lifts around the scenic (dead-end) Berner Oberland, which Rick prefers to a drive-by train experience.

You can also do a shorter scenic train loop via the Golden Pass route - from Bern to Montreux via Spiez and Zweisimmen takes 3.5 hours and the return to Bern via Lausanne can be as short as 1.5 hours.

You are correct with needing to depart Bern at 6:06 with a connection at Visp.

You can verify using the SBB Online timetable/travel planner: http://fahrplan.sbb.ch/bin/query.exe/en

For the given date, enter
From: "Bern, Hauptbahnhof"
To: "St. Moritz"
Via: "Zermatt"

Using 08 May departing at 6:00 as a trial start, the SBB travel planner will offer several routes. You will want to pick the one that includes Travel with "GEX", with GEX the abbreviation for Glacier Express.

Routing displayed is:

Bern, Hauptbahnhof

dep 06:06 Platform 5 InterCity 802 Direction: Brig

Visp

arr 07:02 Platform 6
Change

dep 07:08 Platform 3 Regio 213 Direction: Zermatt

Zermatt
arr 08:14 Platform 3
Change

dep 08:52 Platform 4 GEX 902 Glacier Express 902 Direction: St. Moritz
Reservation compulsory

St. Moritz arr 16:58

In my opinion, it is a waste to go all the way to Zermatt to get to the Glacier Express, then board and come back down the same deep valley where you can't see all that much because it is so narrow. Pick it up in Brig as Laura suggested. If you want to get there on a scenic train, take the "Loetschberger" that leaves Bern at 7:39. There is a 58 wait to change trains in Brig, time for coffee and a strudel.
Schedules at www.sbb.ch/en, and enter "via Kandersteg". Otherwise it will show the faster regular train that goes completely under the Berner Oberland mountains in the Loetschberg Base Tunnel. The Loetschberger goes over the Berner Oberland mountains and uses the shorter old Loetschberg Tunnel. Take a window seat on the right side of the train.

It is a super waste of time! If you want to see the beautiful alps then head for the Lauterbrunnen Valley and surrounding alpine villages of Murren / Wengen / Gimmelwald and just STAY in one of the many hotels there. Instead of spending all those hours on the train looking out the window - just walk out your hotel's front door and gaze at the amazing vistas right at your feet.

I'll stick up for the Glacier Express. It is a pleasant ride, especially if you want to go from one end of the Swiss Alps to the other. The character of the mountains varies a bit along the way. Highlights would be the Oberalp Pass coming out of Andermatt, the upper Rhine gorge between Disentis and Chur, and the "Merry-go-Round" of the Albula railway section with all the spiral tunnels and the iconic Landwasser Viaduct, although you don't see a whole lot of it because you are on it. A headset with multi-lingual voice-over explaining the sights. Nice, although expensive, lunch served at your table.

Thank you to everyone for all the helpful information. I actually feel like I might know what I'm doing now. I am certainly more knowledgeable than I was before I posted. In response to Tim in Knoxville, my husband is a train lover and has always wanted to take a day trip on one. I can't think of a better place to do so than through the Swiss Alps. I guess one person's "waste of time" is another's joy. Once again, thank you to all who responded.

Getting up into those beautiful alpine villages involves riding a plethora of transport options including gondolas, funiculars, trains...it's not difficult to find trains to ride in Switzerland because they're all OVER the place.

Your train lover husband will be interested in the technical details of the Glacier Express: https://en.wikipedia.org/wiki/Glacier_Express

The trip on the Glacier Express is a 7½ hour railway journey across
291 bridges, through 91 tunnels and across the Oberalp Pass on the
highest point at 2,033 m (6,670 ft) in altitude. The entire line is
metre gauge (narrow gauge railway), and large portions of it use a
rack-and-pinion system both for ascending steep grades and to control
the descent of the train on the back side of those grades.

We did the Bernina Express, another Rhaetian Railway operated train and were impressed by the engineering and construction of tunnels, bridges, snow/avalanche shelters and viaducts. An end car is a good seat in that turns are so tight you can see the end of the train from your panoramic window seat.

Thanks for the information. It's very helpful. I'll tell my husband to check out the Wikipedia link and we'll be sure to try to sit in the end car!