of course you can everytime, once your past the maths its the same as any other floor tiling job. you do need to work out of the centre of the room and with a few calculations it is then very straight forward. it would need explaining with drawings as well as text so if you are able to leave your email address i will gladly send you a step-by-step guide for a perfect floor.
Tiles on Diagonal - of course you can. See Dutchie's paper.
No grout? Phew - not a good idea really. Differential expansion and contraction rates 'twixt timber and tile apart from movement of wooden floor means sooner or later something has to give if you don't use flexible adhesive and flexible grout. That's part of the point of it (the grout I mean). Oh and that applies to concrete as well BTW.
You can certainly try but I guess it's probably going to get expensive at some stage.