Ok, agree with all of what everyone else has said that her coming is madness. I emigrated to the US 18where months ago, and I tell you now this kind of scenario would have probably killed DH and I's marriage. unless you deal with it properly.
I get that you are stuck with this, so here are a few ideas, that might help.
Firstly a couple of questions. When do you arrive? I have a wonderful friend who lives in Perth, I'd be glad to hook you up with her if it helps.
OK so practically. if MIL cannot afford a new bed, there are other options. Ebay and Craigslist where you might pick up a cheap one. and freecycle.com where you might find one for free. its perhaps a start. we found loads of stuff that way to help us when we first arrived.
secondly, you really need to set some solid ground rules, for what you are trying to achieve settling /ukrainian routine wise for the kids and you as a family. and come up with some helpful strategies that MIL can use to support DH getting a job, kids settling in and you working when you arrive. Essentially if she is not paying rent or contributing, and is there to help, then she is like a an au pair, albeit an elderly one and you need to have a plan in place to make sure all of you (her included) get something good from it.
My MIL is coming to "help with the baby" when my DS is born next month, she is planning to stay an undefined amount of time. we have negotiated some ideas of how it will work, so we don't kill each other.
Believe me emigrating is stressful, and it can also be depressing at first with home sickness and everything else. so make sure you take control, and steer things how you need them to be.
Best of luck. and PM me if you want me to put you in touch with my friend in Perth.