In my book, a 'straight in' remains an IFR approach, normally accompanied by 'via the IF/FAF' or 'via XYZ VOR' and means you do not need to fly any procedure turns or the like, but you will obviously need to observe safe altitudes. I would also assume that ATC will provide traffic separation AND you will be given descent clearances.
A 'Visual' is just that, a VMC manoeuvre, use whichever beacons you like (or none) but you must be able to complete the approach in VMC.
So, to answer your questions about a 'straight-in':
Can this be a form of a short-cut? YES
If cleared do Crossing Altitudes no longer play a role? NO
Does one have to have the Runway, Approach Lights in sight to request, as I can see no point other wise? NO
Once Cleared - does one have to remain in VMC or maint. contact with above? NO
What about aircraft separation and terrain clearance - will ATC still be required to monitor? YES