1. Is it an expensive job to build on the ruins of an old cottage? (not a large one)
Depends on how "ruined" but in general, it can be quite expensive. You can be lucky and not uncover any unexpected nasties or unlucky and have something unforeseen blow your budget out of the water - dry rot in the roof or significant cracks in walls hidden by inches of whitewash, for example. These kind of things are not always uncovered in advance by an engineer.
There are also many things to address that you mightn't necessarily think of. There are all sorts of issues around the "non-standard" construction of old cottages. For example, they often have very low doors and you have to either raise them or spend your life with a bruised head. You may have to put in extra windows or widen existing ones as the very thick walls tend to keep out light.
2. Would it be cheaper to knock the cottage and build from scratch?
Quite possibly - again depends on just how ruined the cottage is. Also, in terms of upkeep, you may have to put a lot of expense and effort into damp-proofing and insulation and the results may still not be on a par with modern standards. I've seen more than one such cottage that was very difficult and expensive to keep heated.
Expense aside, there is something really nice about a refurbished cottage if it's done well, that you just don't get with a newly-built house, though.
3. Is it easier to get planning permission for a house where a ruin already stands?
I'm not sure but I think it may be, provided anciliary needs such as location of a septic tank (if there isn't already one) can be met within planning regulations. Check with the relevant planning authority.
4. If so, how soon could planning permission come through?
Don't know what the typical waiting time is, if there is such a thing.