One thing that might help with the hoover, as the more she is biting and attacking it the more it will become the norm and learned behaviour is.
Try just wheeling it into the room not saying anything and just leave it there for
an hour or two. Then try to call her over to it, or if she does go to sniff round it of her own accord so much the better and if she doesnt bark or attack it, praise and give her treats. After just try wheeling it backwards and forwards a couple of times with it not switched on. Again if you can get her near to it without barking and attacking, then praise and treat. Next step, switch it on and off immediately, praise and treat if she remains quiet. Do that a few times.
Next step let it run a little longer, then once or twice backwards and forwards with it running. No guarantees of course and it may take a few days if not longer and you will have to do each step at her pace. Worth a try though.

If you do have success with the hoover then you can progress to the snow shovel I think it was you said. However on reflection it might be better to start with the snow shovel first as its not so noisey and large. If you dont mind, you could bring that inside and just park it somewhere like I suggested with the hoover, and do that first and the hoover 2nd. Basically the idea is to gradully desensitise her to them and make a positive association with them rather than
the negative one she has now.

One last thing about the trading/swapping things. You need to aim low and work up higher to get her to take in the concept. Start with more boring uninteresting things that she doesnt highly prize to trade with high value treats she really likes,
maybe chicken. Offer the chicken and if she drops what she has then
Say drop and swap with the treat. She should learn the verbal command drop then also. As she gets better with lower value things to her than you can work up.

Hope these further suggestions will be of some help.

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Have you thought about crating the children, so that they can be kept out of the dog's way unless you can TOTALLY supervise interaction?

I have thought about it and it's a good idea, but you know crates cost a lot of money. And right now I'm reserving all of our available cash for the dog. The kids have already been complaining about how all I feed them is spaghetti and Biscuit gets to have their meatballs. I think it would be cheaper to just put them in their closets. Then again, I'd feel bad because Biscuit doesn't have her own closet. Better to put both of them in ONE closet and then she can have the other.

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Thank you for the advice Dodigna and I will put both books on my list. 'Mine' sounds like it would be quite helpful and I'm very interested in the clicker training. I'd like to start that out at home and then book a course for us to go to. And no, I didn't feel you were too harsh at all, I know you are all very passionate about animals on this board and appreciate your being honest. I can't rep anyone else till tonight (I didn't realize there was a limit, I'd just found out what the scales mean!), but thanks again.

I will try out your suggestions as well Sled Dog, it would make life easier. Biscuit got quite used to the hair dryer, blender and my kitchenaid, but they don't move around the carpet or sidewalk. Thank you!

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I managed to stop Kilo's barking and general craziness whenever I hoovered using the method described by Sled dog hotel; it worked surprisingly quickly. I did in fact hoover tonight with no dramas

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LOL!

You know, electric collars get a lot of criticism but I've heard that for some children they can work wonders

Got my order in! I'm getting 'Click to Calm', 'Mine' and 'On Talking Terms with Dogs'. That should be a good start. I've also been reading all the clicker threads looking for insight, thanks to those who post them. Progress is slow with the vacuum, she'll approach it and sniff, but as soon as I move it - bang - and it hasn't been on. I'll keep trying with that or stop vacuuming!

Our son has improved although we still have to stop him at times - he is an impulsive child. The kids and myself have also been practicing dog signals on each other (not Biscuit) - whale/half moon eyes, freezing, licking of the lips etc.. We're a bit frucking nuts in this house I think.

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