Hi Danielle,
OK, wow, you have a lot going on here. For future reference, get the leases, payment history, etc during the purchase process.
I bought a property once that where the sellers family was one of the tenants, and it was rented below market. Regarding the comment to not renew them: I get it, and in your shoes I'd probably rather be rid of them, but would stop short of a hard line. There could be some good tenants here. In my case, though I was initially very worried and cautious, the remaining family member turned out to be a great tenant.
Month to month is not a bad spot to be in, given the circumstances. Check your state's rules, but with month to month you could literally increase the rent as much as you want with 30 days notice. When a lease is over, it's typically easier to get possession if things go south than just a full on lease dispute. Ie: if the lease (including m2m with proper notice) is expired, there's little to dispute.) Again, check your local regulations, but you could theoretically get those rents up very fast. And similarly, if the applications reveal liabilities, you can simply give notice on the m2m.
First, get as organized as possible and present yourself as a business, even if it's just you running a business. In my experience, they react differently to somebody that's part of a business than "Danielle my landlord." You can use language like, "These are the policies our company follows." Be professional, polite, but firm. I think it's reasonable to request the full app if you're going to consider continuing the leases. Provide the applications and tell them "they are due with" (not, "I'd like them by") next month's rent.
It might be hard to get that tenant off the 15th, especially if that's part of the formal agreement they had. How important is that? I'd give them some time to catch up. Of course, again, with the m2m, you have a strong position. You can non-renew if they won't play ball. "Our ownership group doesn't give any leeway on payments being due on the 1st. We realize this will be a change for you. What can we do to help get this resolved?"
I would go hard at the late tenant. 1. Send notice the rent is late. 2. find out the maximum legal actions you can take and go. For example, where I live in IL, there's an automatic 5 day grace period before a fee can be charged. But on the 6th, I would charge the maximum fees. Find out also about any requirements for a 5 day notice and get prepared for that. Again, you can probably terminate the lease with a 30 day notice. If they pay up, you might let them know that future late payments will result in non-renewal of the m2m lease.
For the rent increases: doesn't have to be all at once. I had a property that was way below market rent. I increased it $50 every 3 months until it was back to where it needed to be. Tenant would have preferred to keep paying the lower rental amount, but she stayed.
It may be worth it to discuss with each tenant face to face. You can learn a lot about them. I try to give the benefit of the doubt, although I stick to my rules and policies. There are plenty of hustlers out there, and you have to protect yourself. At the same time though, there are good renters out there, and who can blame these people for getting cheap housing from a family member? If you are professional and firm, the ones that could've afforded more all along will probably pay and the others will move on. If you come across too heavy handed, they could unite against you. I still think you're in a good spot to resolve this quickly, but better to avoid a conflict.
Good luck. I'd be interested to hear how it works out. Always more to learn!
