Presumably there is nothing stopping the 395 being hauled through the tunnel by an approved locomotive (like the ICE3 etc). Obviously this isn't suitable for a normal passenger service.
Certainly not. The majority of new build rolling stock for Britain gets delivered in the consists of freight services heading through the tunnel.
It's worth considering the 395s have very good acceleration so would be able to get up to 140mph after leaving the tunnel quicker than the E*. Perhaps it would be possible to build a loop and a platform (built to UK loading gauge) in Calais for such a scheme.
If they are to stop at either Calais itself or the Calais-Frethun TGV station outside town there would be no need to accelerate rapidly coming out of the tunnel.
Where they went (if anywhere) after Calais would determine whether high speed running would be required, and whether 225 km/h would be enough. Were British-gauge trains to be used, terminating at Calais and having passengers transfer to SNCF services might be the best bet.
My opinion is that it would be better to operate a cross-channel stopping service as an extension of the Lille-Calais TER-GV services than as an extension of the British network. Such a move would allow it to happen with very minor modifications on the British side (is Ashford fit for UIC trains?) and none on the French side.
Actually loading gauge might very well be a problem as 395s don't have retractable 3rd rail shoes and the UIC loading gauge could be smaller where the shoes are depending on the line.
No, the on-board equipment for British third rail use comfortably fits inside all the UIC GA, GB, GB+ and GC gauges, plus the German G1 and G2 gauges. If it didn't, British third rail stock would not have been able to complete testing on the Siemens track at Wegberg.