Agreed on the limo bus option to the hotel in Tokyo, especially if you have luggage.

The JR Pass is usually a good deal, and if I remember correctly, the price for a 7 day pass is already cheaper than a round trip ticket from Tokyo to Kyoto. You can also use the pass on JR trains in the city (e.g. Yamanote line).

I am of the opinion that Mt. Fuji looks better from a distance than up close, especially when there is no snow. It's a volcanic mountain, so there isn't much vegetation on the mountain itself. Probably better, as suggested above, to go somewhere with a nice view from a distance. The five lakes are a popular place to see Mt. Fuji, as is Hakone. But being able to see it well depends a lot on the weather.

You should be fine with the luggage in green cars. Ordinary cars are a bit crowded, but there should be enough room in green cars either in the overhead spaces for smaller bags, or just by your legs. The last seats usually have a space behind them that can take a lot of luggage so that might be an option as well.