Well, if she lets you have the house and she was the one to get an apartment, her living expenses would be a lot more manageable for her, wouldn't they! But be tactful if you point that out, of course.
You already have joint custody by default. That just means you and your ex both make the decisions for education, health and religion. Don't give that up!
What you want is shared access. The children living with you half the time, and your ex the other half. At least for the younger children; sounds like your oldest isn't willing to be shared. Then wanting to hang out at your place, and you taking them to all their sporting events will help you make your case, but the biggest issue will be where do they sleep? Having them sleep at your place half the time right from the start, even if it's crowded in your apartment, will be a big determinant. And as you described, you have 3 kids at half time, and one kid full time, that gives you a slight majority as far as I'm concerned, and you should definitely fight to be the one who keeps the house (if you can afford it solo, of course). If you want to be forceful and call her bluff, apply to the courts for exclusive possession of the matrimonial home. Don't delay, as you want to be establishing status quo from the get-go, and the kids would be a lot more comfortable in their own beds anyways. If you want more discussion with the ex, just make sure the kids sleep at your place half the time until you come to an agreement about the family home and get the funds you need for a bigger place at the very least. Read up on here, don't just ask for advice, but read past threads for lots of similar advice already posted on how to orient your arguments for shared access. An essential thing to make clear in any agreement is that you are not trying to keep the children from their mother.
And by the way, more part-time jobs to make more income to pay the CS just raises your CS and uses up time that could be spent with the kids. It's definitely a diminishing return on time invested.
If she goes after your government pension, go after hers. Call her bluff. But do it out of 'fairness' not aggression.
With the limited information you've provided, it sounds like your ex is basically just trying to manipulate you. She's getting CS from you by playing the "oh I need food" card and using threats to avoid you going after your fair share of the matrimonial home. Do your due diligence research here to avoid falling for any tricks. And hopefully soon someone more familiar will be able to calculate a true CS amount for you, or you may want to consult a lawyer, but I don't think, if the access is currently shared for 3 and in your favour for the oldest, that you should be paying as much as $800. If you aren't sharing the 3, get them sleeping at your place half the time, or get back into the house with them one way or another, as you don't want to give her time to establish status quo of them not being shared. You sound like an involved father, so stay that way! Your first post seemed quite shell-shocked, but you need to get your head back on straight as soon as you can, armed with knowledge, to give your kids the best dad they can have during this stressful time. You feel stressed out and at least you have a modicum of control over the situation. Their lives are crashing down and they can only watch.