*First of all; brands to buy and brands to avoid, for both batteries and charger?
Turnigy are good for both, Nuprol are shit for both (well...pretty much everything in fact)
*I see that some come with Deans connectors and some with Mini Tamiya - are the Tamiya ones ok to use or do I have to rewire all of my AEGs? Or use Deans to Tamiya adaptors?
Mini Tamiya are perfectly serviceable, Deans are in theory better but it does depend on the quality of the ones you buy and someone will be along in a minute to tell us all that XT60 are in fact God Tier and everyone should have them.
*Is the risk of fire and/or catastrophic failure exaggerated under normal/sensible use? Do I really have to keep them in an asbestos bin at the bottom of the garden?
Be sensible, use a decent charger and keep them in a LiPo bag. Catastrophic failure is almost always a result of neglect or abuse. Puffing batteries is something to watch out for but if you follow the earlier points then you should be fine. If you notice any getting physically damaged or puffing up then just get rid and replace them, they're really not expensive.
*Most of the impetus behind this is that I'd like to be able to get a full day out of one set of batteries (at the moment I'm only getting a bit more than half a day out of stock NiMhs), is this realistic?
Entirely depends on how much battery space you have and your playing style. That said, the stock NiMhs that come with guns are almost entirely dogshit.
*So far all internals for my AEGs are stock, so I was going to go with 7.4v, perhaps upgrading to 11.1v if I upgrade gun internals later on - is this sensible?
Most modern AEGs are fine with 11.1v but lifespan may suffer a bit if you're particularly ham fisted. Older guns would be better on 7.4 until you can beef them up
*Any reason not to get the biggest/highest mAh I can fit into the gun?
*Anything else I should know getting into this?
Always check the sizes of where you plan to put your battery in your gun BEFORE buying the battery. Just because the website says it'll fit in an M4 stock tube, doesn't mean it'll always fit YOUR M4 stock tube.