In Europe, practices vary from country to country. In France, the trains run on the left, though I believe they run on the right on the Paris Metro.
In France trains run on the left except in Alsace-Lorraine where they run on the right. This is because Alsace-Lorraine was part of Germany when the railways were built there, and German trains run nominally on the right side.
Most lines on Paris Metro run on the right but there are a few exceptions. The RER lines run on the left since they connect to the main line network.
With all of this said, however, however, I believe most major American railroads (including Amtrak) can operate their trains on either the right or left track, depending on which is safest or most convenient in a particular location. I think that Amtrak trains are often moved onto the track closest to the next station, especially if (as in many small towns) passengers would otherwise have to walk across a track when getting on or off the train. So long as signals and signs can be read by the engineer of a train running on the "wrong" track, there really should be no problem.
If the signaling system is setup to allow bi-directional signaling then trains can run on whichever track. If they are not, then it takes a track warrant (e.g. a Form-D under NORAC governed operations) to do wrong track running.
For example on the NEC in 4 track areas between Newark NJ (CP Dock West) and Marcus Hook PA (CP Hook), the outer tracks are signaled for unidirectional running (NORAC Rule 251), except around stations like Metropark, Trenton etc. The inner tracks are signaled for bi-directional running (NORAC Rule 261). Due to track outage if a west bound train has to be moved to track 3 from track 2 that can be done easily since track 3 is Rule 261, but to move it to track 4 takes a Form-D.
So bottom line is it depends on what signaling rules apply on which track in that territory and what authorization the train is carrying at the moment. This is uniformly true in all countries with significant rail infrastructure, though of course the rule books that apply do differ, and track authority conventions and protocols also differ significantly from country to country and even from region to region (e.g. NORAC and GCOR in the US) and even railroad to railroad (e.g LIRR rules differ from NORAC, though they do use NORAC Harold to Penn Station).
As for train operating position, the tendency on high speed trains is to place the operating position more or less in the center. Interestingly, even though the French LGV (and British HS-1) do not have any trackside signals at all in TVM 430 (or 300) territory, they do have trackside block marker boards (
Repere Board) which are always on the left of the track. I wonder what they will do if they ever build any right hand running LGV