I don't want to register each in my name and wait 2-3 weeks for the new V5 to come through before I can sell it, so if I register the car to my business (i.e filling out the little slip that says you're selling it to a trader and therefore the car is still in the previous owner's name), can I still add the car to my personal policy for a day or week?

Well yes i suppose you could do that. All the 'sold into trade' bit of the V5 really does is mark the vehicle as 'in trade' at the DVLA so its treated the same as being Sorned as far as tax is concerned (if theres no tax on the car). Plenty of 'off the books' home traders try this and come unstuck later as the Inland Revenue and DVLA are not stupid, they can find out if you're really a trader or not but you've already set up a business and it'll be registered to that business i presume.

Read your insurance terms and conditions thoroughly, the vehicle would still be in someone elses name obviously and some companies wont do it. A trade policy covers you for things a standard insurance policy doesnt, so you'll have to read all the small print and talk to your insurance company, if you have an accident and they find out its a business use in-trade car when they think its your personal vehicle they wont be happy. If you do insure it on a personal policy make sure its covered for business use. Im not sure how you intend to obtain the cars in the first place, most auction houses will require proof of being a trader (ie your trader insurance policy certificate) or they'll fill the V5 out for you as a private buyer anyway.