@Meghan Sullivan first of all welcome to BP from a fellow western New Yorker. Next as it seems to be the most popular question you asked is regarding the security deposit I wanted to throw in my 2 cents. I agree with @Kevin Siedlecki that you should be collecting a security deposit if nothing is in place now. But I would confirm with the tenants that they did not pay a security deposit. It would be terrible for you to take the sellers word then send a letter to your new tenants requesting one and they tell you they paid one.
In response to the question on rent increase, you did not say what the current rents are so its had to give you my full input but $100 a month seems like a heavy increase right away. I know you said you would not mind if one tenant left so you can rehab a unit but a large increase right away might scare both of them off, and as you said you will be going into the holidays and the cold months in this area when its hard to find new tenants. Just think about the money lost with a vacant unit compared to the unit being undervalued by $100 a month. Something I just did with inherited tenants that were undervalued on rent and month to month, have them sign a new month to month agreement right away with a small increase right away maybe $30-50. Then keep them on month to month till spring and if the tenants seem responsible offer them some options (1 year least with an increase of $20-30 a 6 month lease with an increase of $30-40 or if they want to stay month to month an increase of $40-50 more.) if the tenants are good and willing to stay long term you are better off with small increasing and keeping them in place then losing them and having to rerent the unit.
As far as updates when either unit is empty be careful not to spend money on updates that will not allow you to raise your rent. You have to avoid the urge to make improvements because it is something you would want, you are not living in this house you want to maximize your return on any improvement.
When it comes to a business being run out of the unit, you need to put an end to it right away, it the contract and actually verbally notifying the tenants, just because it is in the rental agreement even if someone falls that rental paperwork will not stop a lawyer from filing a lawsuit against you.
And finally in response to the lawn cutting, you can ask the tenant to take on the responsibility and discuss with them some kind of compensation if they are interested in doing the work. But I personally would not just automatically put this on a tenant, part of the reason some people rent in the first place is to avoid these responsibilities. It is ultimately your responsibility and you don't want to lose a good tenant or scare away future tenants with lawn care that you can hire out.