Swiss Travel Pass :Zurich -Interlaken-Lucern

If you have a Swiss pass it is valid for all transport in Switzerland, rail + boat + bus + city transport, except some mountain railways and cable cars, but you usually get a discount for these.
There is a map (which comes with the pass) which shows what is covered. Here: https://www.sbb.ch/content/dam/sbb/de/infotexte/uebersichtskarte-sts.pdf
Every train route in Switzerland (every red line on the map) has at least one train every hour.

Q1: Yes, you can travel on any Swiss train, anywhere to anywhere. IC and EC trains are just normal trains, nothing extra, and are covered by normal tickets or passes.
"October 26 - Interlaken - Lauterbrunnen - Murren - Schilthorn" - OK as far as Mürren. Above Mürren it is a mountain cable car, marked in grey on the map, not covered. You need to buy tickets at Mürren.
"October 27 - Interlaken - Zermatt - Gornergrat and return" - OK to Zermatt, discount above Zermatt. Again see map.

Q2: Again yes, Interlaken to Zermatt is covered.
"October 28 - Interlaken - Lucerne - Mt. Rigi - Lake lucerne." - Rigi IS covered, see map.

Q3: Take a boat trip on the lake when you are in Luzern. Boats are covered as well.

With a Swiss pass, you just get on any train, nothing more needed.
October is not a usual time to visit the Swiss mountains, the summer season is over, and the ski season has not yet started. You may find some restaurants and tourist shops closed. There will be snow at higher altitudes.

This year, the Swiss Travel Pass is valid on the Schilthornbahn all the way to the top.
https://schilthorn.ch/cmsfiles/tariffs_schilthorn_abjuni_2018_english.pdf
Read this line in the Schilthornbahn price list in the above document.

The Swiss Travel Pass is valid on the entire journey Stechelberg – Mürren – Schilthorn.
On the funicular Mürren – Allmendhubel the “GA travelcard” tariff applies.

The Gornergratbahn you get a 50% discount. However, it would be a waste if the weather is not clear when you arrive in Zermatt. I guess you can make your decision on buying a ticket when you arrive in Zermatt, or even if it is worth going to Zermatt that day.

Thanks Chris and Sam!

I understand that Swiss Travel Pass is valid

1. Zurich to Bern to Interlaken
2. Schilthornbahn all the way to the top.

In case of the weather not being clear, how can I spend time in Zermatt?
Or is there an alternative to my itinerary on that day!?

Zermatt is basically the mountain and some exceedingly high priced (even for Switzerland - if you have to know the price you can't afford it kind of priced) stores. That's it. Period.

If the weather in Zermatt isn't excellent there is no need to go.

Many museums are covered by the pass. There is a nice, small art museum in Lucerne that is covered. The Lucerne transportation museum in not covered, however. Are you staying in Interlaken? I would stay in Lauterbrunnen, it is far prettier.

There are waterfalls in the Lauterbrunnen Valley that are nice and could be visited on overcast days.

And boat trips on Lake Brienz or Lake Thun. Or walking around Thun. Though all of those experiences would be more pleasant in nice weather. Nice weather, however, is not even guaranteed in mid-summer.