Road cycling route questions for Lake Brienz, Meiringen, Grosse Sheidegg, Grindelwald Loop

The trains in the area take bikes. You need a ticket for your bike, but not a reservation. You just load your bike in the designated spots. From Gundlischwand just ride your bike to Zweilütschinen, and board the train to Grindelwald there.

Note that you can find all public transport routes, and wether or not they take bikes, and under what condition on www.sbb.ch, so you can start your research there.

Is there a bus/train that takes bikes from Meiringen to Zwirgi?

This is the Post bus route. You can read about bike reservations here:

https://www.postauto.ch/en/leisure-offers/excursion-tips/rosenlaui-route

I took this bus about at the beginning of July, did not have my bike, took one of the first departures of the day, no bikes were on the bus at that time and I think we only passed one or two on the road all the way over Grosse Scheidegg and down to Grindelwald.

This is a very popular route for cyclists though and the bus driver said that there are usually a lot on the road.

Post Bus runs between Meiringen and Schwarzwaldalp. Grindelwald Bus runs Schwarzwaldalp - Grindelwald.

https://www.grindelwaldbus.ch/en/timetables/timetablesummer/mountain-lines/line-128.php

Is it preferred to go East to West or West to East Between Grindelwald and Reichenbach falls?

Can't speak to this since I did it from the comfort of the bus with no exertion at all ;-), but just want to add that for sure you want to catch the view around the Gschwandtenmaad bus stop where the river and mountain line up for iconic photos, so if you happen to be cycling downhill at that point, turn around and take a look.