Interesting your two options are to either run and hide or to try and be even nicer than you have been.
I'd say neither option is a good one.
If you are trying too hard to bed over backward for someone who is treating you like a doormat, you will not get anywhere. In fact, you will just cement her resolve to trample all over you even more.
In a situation like this, a partner (your husband) will not see what is going on and it is futile to try and make them. This doesn't mean you should 'put up and shut up' because you shouldn't; but it does mean that while you are within your rights to verbalise what is happening, you would be wasting your time to expect him to see it or agree.
These men are conditioned to accept the innocence of their family without question, and they don't want to rock the boat with their own flesh and blood, so you cop it instead.
I would be interested to know what your husband thinks of the accidents and safety aspect with his sister? I assume he thinks you're being overly protective and that she is being misunderstood?
There are two ways to look at it. On the one hand, is there any possibility she was ignorant about the safety of the internet and was tying too hard to be a cool aunty? Is there any chance the table lamp incident could have shocked them all and make it certain never to be repeated as they've learned from their mistakes? Only you know if this is a possibility. Or maybe she is just a walking disaster that should be kept away from childminding!
So often it comes down to one parent having a better instinct for the safety of children, and it may be it is always going to be your job. Trust your instinct though, as you'd never forgive yourself if a major incident took place and you had ignored your inner alarm bells. You can't expect your husband to share that view though, nice though it would be.
If anyone comes to your home and belittles you or is unpleasant in any way, you should not try to be nicer! You are not standing up to the bully and will accomplish nothing. But you don't have to go all guns blazing either, because as you said yourself, it makes you look like the trouble maker.
So - it's hard to advise when I don't know what she says to you, but I do know that there would be some people out there who would be affected differently by her. Maybe not so intimidated. It reminds me of the saying that 'people can't make you react a certain way; you have a choice in that yourself. Sometimes that choice has to be learned through practice though.
So if she is putting you down, challenge her in a neutral tone (not friendly, you need to be genuine, but not fighting talk either because you'll look like the baddie unnecessarily).
Verbalise what she's doing. Ask her if she's ok "you don't seem too happy today, are you ok?". That can take the wind out of someone's sails.
Regarding the children. Every time she asks you can ignore the fact she is a broken record and use an age old assertiveness trick - be a broken record back. So when she asks "can I babysit the kids tomorrow" you can say "no, sorry. I'm afraid that's not possible. But you're welcome to come and visit/meet out on x date instead if you like?"
Then walk away and don't expect her to be gracious. But carry on in your own space and work hard at being unaffected. Don't doubt yourself (her undermining you is for this exact purpose). Do something nice for yourself, have a positive though or experience. Find a way to rise above it.
If she sees no one is reacting and she is getting nowhere, there will be no advantage to her maliciousness. She only does it because it DOES affect you!
Removing yourself from her company is a different matter. There is a fine line between choosing not to be around toxic people and running away from them. I'd say it will sent out the wrong message if she can send you off into the the other room as though you've been dismissed!
Try not to dread her visit. If you rise above it then you will be able to feel neutral about your husbands poor twisted sister who sometimes visits!