Battery, alternator, loose connection? HELP!!!

Hello, right I apologise in advance for a long post but please bare with me.

Right the supra has been off the road for just over a year due to having A baby/moving house etc etc, before I took it off the road we where already having battery issues, wouldn't start when cold or it had been sat a while, I just assumed the battery was on its way out as I've have the car 3 years and hadn't changed the battery and didn't know when it was last changed.

Whilst it's been in storage I've been using jump leads to start it every now and again and then running it up to operating temperature etc. About 2 months ago we started the car using jump leads but when we removed them it kept cutting out, on closer inspection of the battery we noticed white powder leaking from it, so I brought a new battery.

We put the new battery in the car but it wouldn't start (when you buy a new battery do they have charge?) so we started it using the jump leads again, when I removed the leads it cut out again, we used the leads again, kept them on for longer and I kept my foot on the gas whilst we took the leads off, It kept running until I took my foot off the gas then it cut out again, I finally got it idling by holding my food on the gas for about 15 mins then gradually released, I just assumed this was because it hadn't been started for a while and everything just needed to circulate.

We haven't been able to get over to start the car for various reasons until yesterday/today. The plan was to go over yesterday start it up, check all the lights etc etc for the MOT today. Started fine with the jump leads yesterday and ran after taking them off for a couple of mins then died. Me being nieve thought it just needed a run to get the battery charged. So today we jumped it, fine, drove to the mot centre (about 10 miles) fine. Left the car with them.

Got a phone call from the MOT centre saying they couldn't carry on with the mot and it was stuck on their ramp, they had had to use a battery pack to start it but it still kept cutting out and it had drained their battery pack :/ so they had to charge that before being able to use it to start the car again. He told us to come back in an hour once he'd got the car started and moved. So we went to pick it up, matey at the garage said he thought the battery was knackard (but it's a new battery) anyway we had to jump it to start it, drove away, drove for about half a mile lost all dash lights, dials and powersteering, but car still ran for about a mile before cutting out, had to jump it again, same happend but managed to get it on my parents driveway before it cut out.

Took the battery out and plugged it into the trickle charger, expected it to say it was compleatly dead, but it was showing up as half full??? Please help, any suggestions???

The leads to the battery or the connectors are loose/damaged etc if you have a multimeter you should be able to work out which is at fault by testing different parts and checking the voltage...

Ok thanks, I'll get my dad on the case in the morning, I'm sure he'll have a multimeter.