As I said in my previous post, the only legal restriction on tyres of different types on the same axle is that radial ply and cross ply tyres must not be mixed on the same axle.
Regarding a new tyre on one end and a worn tyre on the other end of an axle, there shouldn't be any strain on the diff, if any. However, it might be a reasonable idea to try to balance out the wear around the vehicle, for instance put the new tyre on the back and then pair it up with the least worn of the other three, the theory being that the grip of a new tyre and a very worn tyre on the same axle might have a detrimental effect on the expected handling, particularly in wet conditions.
Everybody else have had their say about what's good practise or best practise, and yes the best way for you to go would be to try to replace your tyres two at a time, those of us fortunate enough to be able to afford to do that often forget about the days when we struggled to put new rubber on a vehicle.