ugh, so many questions...so poorly organized. XD
As I understand it the pokemon offspring will inherit TMs from dad, level-up moves from mum & dad and egg moves...I don't quite understand, you don't fight with an egg do you?
No, you don't fight with an egg. You walk around with it until it hatches (just like the event Togepi egg you had at the beginning of the game).
The baby's base moves will be their species' base moves plus any TMs/HMs the father knows that be baby is compatible with, any egg moves that the father knows, and any level-up moves that both parents know. If all these amount to more than four attacks, the base attacks will start being eliminated (because they're easily relearned with a Heart Scale)
Also, the pokemon you wish to end up with needs to be able to obtain the moves you want ordinarily anyway. So, if you want it to inherit a TM from dad the baby would have to be able to learn that anyway. I see the advantage in this i.e. you've only got one of that TM but level up moves I don't quite understand...if the baby you want would ordinarily be able to learn those moves why do you require breeding to pass them on?
For TMs, it's as you said, to get more use out of one TM, but for level-up moves, it's usually to get a baby knowing a very powerful STAB move as soon as they hatch, i.e. a Skorupi born knowing Cross Poison.
More important than TMs and level-up moves are the egg moves that can't be acquired any other way than by breeding the attack on the baby. :P
And chain breeding...what are you trying to achieve? lots of TMs in the same pokemon?
Chain breeding is passing a TM/egg move from one species to another by way of overlapping egg groups. For the sake of simplicity, I'll give an example with a universal TM, Toxic. You can use your precious TM06 on a male Snorlax and then breed it with a female Venusaur and the (preferably male) baby Bulbasaur will know Toxic. Because Bulbasaur is in two different egg groups, it can then be bred with say, a female Cacturne so that the baby Cacnea knows Toxic. Since Cacnea is also in two egg groups, it can then proceed to breed with a fairy, and so on...
My idea was to breed a Corsola that was quite powerful but I don't really understand the whys and hows of the method involved.
What you're describing seems more like breeding to get perfect IVs and I know nothing of IV breeding other than it takes extreme patience. XD