Do they have a feature for renters to pay online (ach, debit, credit, ?)

The prepaid credit cards.

Auto-debit, or they can put the money in the bank directly.

While the national ACH system is regulated by the Fed Reserve, soon with Cozy you'll be able to accept credit cards, and even push the payments to a debit card - which should shorten delivery to 1-2 days.  

You should just receive the security deposit + pre-paid rent (one month's rent), along with the lease signing.(2) Great - they should sign the lease with the start date + pay the deposit + pay pre-paid rent before you take the property off the market.

Not sure on this one but an alternative is to get a debit card for your business account.

Seems as a tenant I would be less comfortable giving direct access to debit my accounts than to use other forms of payment.

Have you heard that they eliminated cash in Sweden and you are required to use your plastic credit/debit cards for everything in an effort to keep track of where you have money so they can get all the taxes possible?

A landlord shall not demand or receive security, however denominated, including, but not limited to, prepaid rent in an amount or value in excess of one and one-half month's rent.

As to trivial expenses showing on the bank statement, it could be a red flag, but with the rise of the debit card and move by some to go cashless, it might be OK.