Hi, Kevin!
I've only seen a few re-barrelled .303 Enfield rifles, and have never shot one. The aussies are famed for re-chambering and rebarrelling the .303 Lee-Enfield actions, probably for economy reasons. The wildcat calibers used (.17-303, etc) are downloaded because of the limited action strength.
First, are you referring to the Lee-Enfield design, or the P-14 Enfield (a mauser type) action? The P-14 is definitely much stronger and suitable to rebarreling to another caliber.
The .45-70 rebarrelled No. 4 rifles are practical because the WWII era No. 4 rifles are the strongest of the Lee-Enfields available.
As for the safety of firing a .30-40 Krag round in a .303 rifle. YES, it can be done, and at relatively little risk (the .30-40 bullet is smaller diameter than the .303 bore). However, this would strictly be an emergency thing to do, and not otherwise.
The reverse, firing a .303 cartridge in a .30-40 rifle is not possible. The .303 case has a longer head-to-shoulder dimension and will not fit. Even if it did fit, it would not be a good idea because the .303 bullet is oversize for the .30-40 bore.
As for rebarrelling a P-14 to 7mm magnum or other magnum calibers, it is one of the best and strongest actions for that purpose. Conversions to .30 Magnum, .338, .375, .458, etc are common.
HOWEVER, the .303 Lee-Enfield is NOT considered a good choice for rebarrelling to another caliber. The only possible rechambering I've seen or heard of is to the 7.62x39 Soviet cartridge, since both use the same bullet diameter (.311 inch).
So, there you have it: P-14 okay for rebarrelling or rechambering to magnum calibers, but not the .303 Lee-Enfield. Lee-Enfield okay for conversion to 7.62x39 Soviet only.