Is it a good idea to route tracks via the pads of N/C pins of components to gain access to other pads from the inside of the chip footprint?
Completely depends on the component. Most components just say "NC", "no connect", and I leave those alone. Some say "DNC", "do not connect", and you
have to leave those alone. Some say "NIC", "no internal connection", and you can do whatever you want with them (within reason, of course; no component not rated for it is going to like 1000V appearing on a random pin). And some components flat out don't tell you the whole story in their datasheets. For example, the AD9257 has two "DNC" pins, which should mean they have to be left alone. But ADI engineers say it's OK if they're grounded, as they were on the earlier, otherwise-pin-compatible AD9222. So sometimes you can ground NC pins, sometimes you can't.
I just leave the damned things alone, and it's never served me wrong.
Can tracks attach to any edge or corner of a pad or should it always be 'axial' to the component?
Traces coming out straight from the center of a pad are best, but there's no hard and fast rules. SMT is pretty forgiving when it comes to this, at least at larger pad pitches. The smaller things get, the more careful you must be. I have had no trouble with all sorts of crazy trace routing on 0603 component pads.
Is it good/bad practice to route tracks between pads and under components as I would do with say a axial lead resistor or between rows of DIP pins?
This is OK (in general) if and only if you have solder mask. With solder mask, you can run a trace between the pads of an 0603 resistor with no trouble. Without solder mask, I think trying to do that would likely end in frustration. As others have said, this is a very good incentive to get professionally fabricated prototype PCBs. It's so cheap these days that there's little reason to bother doing it yourself.
Is it OK to lay components at 45 degrees to others?
Sure. Modern pick and place machines have no problem with that (as long as the usual mechanical clearances are maintained, of course).