@Nuo Shi purchasing a property with existing tenants certainly has it's risks, but it seems like you are asking a lot of the right questions to mitigate that risk.
As it has been stated many times before, existing leases come with property, so what are the terms of the existing leases? Month to month or fixed terms? And when do the leases expire?
1. You can background check and credit check existing tenants.
2. If each estoppel cert is signed by each of the residents in each unit, then it seems like there should not be too much of a concern there. You should not reach out to the tenants directly unless the seller gives you permission to do so. Likely, this process will be coordinated through your agent and the sellers agent or your attorney and sellers attorney. This blog post will help you confirm all the necessary information is included in the estoppel certs provided by the seller: https://www.biggerpockets.com/...
3. You may ask the tenants to sign a new lease, but they may or may not agree to do so. If they agree to it, then go for it. You do want to have a 'lease assignment agreement' completed to transfer the current lease to you as the new landlord.
4. You can ask, but the tenants may also not provide that if they are under an existing lease. You should ask the seller for the 'current rent roll' for the past 2 years for each unit with existing tenants to understand when the rents were due and when the rents were collected. This will indicate whether or not these existing tenants are paying every month and paying on time. You then should talk to them after you close to understand their personal situations.
If it has not already been provided, then ask for the lease to each unit to compare with the estoppel certs. You can always have your agent or attorney ask the seller if you can talk with each of the tenants before closing, ideally this would without the seller present.
Hope this helps and best of luck during the due diligence process of the 4-plex.