If you look at the state by state maps, it looks as if there are basically no actual plans to build anything in Wyoming other than an existing short segment around Casper. More generally, rail through mountain ranges made use of scarce passes and built expensive tunnels and bridges. There are some examples of abandoned route segments that can still be walked (and maybe biked) on in whole or in part--for example near Steven's Pass in the Cascades. But I imagine the main routes through major mountain ranges are still in use. There are gaps in Montana too and a small one in Washington.
I assume it's the same reason people bike on I-5. If there's a good route through the mountains, either a major road or rail or both is likely using it.
The better example of an abandoned route through the Cascades is the old railway route through Snoqualmie pass, which connects the metro Seattle area to the east side of the mountains. It's now a state park, and you can hike or bike from one side to the other.
As technology improved, a longer tunnel was built lower down the pass to decrease the elevation gain for trains, and the old route became obsolete. I suspect that this is not a unique story, and the question would be whether the old railway routes are still safely navigable.
This happens a lot. But the new road/rail often shares the line of the old on the easy sections, so you may not get a continuous bike trail out of the old route. (Plus, as you say, old bridges/tunnels/steep slopes may be unsafe, or simply damaged & not repaired.)
The Eisenhower Tunnel is just an automotive tunnel. I believe rail always used the Moffat Tunnel since it was completed in 1928. The two aren't that far apart. (AFAIK it used the Rollins Pass--which had problems with grade/avalanche/etc. previously or just bypassed to the north or south.)
To a very large extent, that's probably true, but there are some situations where a route that's OK for one mode of transportation is unusable for another.
Trains need gradual inclines and really large radius turns. Cars can handle steeper inclines and sharper turns. If necessary, humans can do very steep inclines and very sharp turns. And they can handle narrow paths. You'd obviously want to minimize that, but it's possible.
Since the requirements are different, the possible routes may be different.
Fair enough although I assume that realistically a bike is going to want at least a 3-season 4WD road through the mountains--and even that may not really be suitable for someone doing distance cycling. And, to your point as well, there are certainly (usually 3-season) roads through mountain ranges that may not even be especially challenging but aren't suitable for high volume traffic.
There are also rare situations trains can handle, which road vehicles can't (safely): very long tunnels, extreme cross-winds, limited width for the track/road, steep ascent (rack railway).
I assume it's the same reason people bike on I-5. If there's a good route through the mountains, either a major road or rail or both is likely using it.