He is a good worker, but I am very concerned about having him work on occupied properties (so far I have only had him do exterior yard work on vacant houses)--   When I confronted him  about what we found, he said that I was happy with his work so why am I concerned about lhis past & doesn't everyone deserve a chance, etc.What would other landlords do? 

If you uncover any concerning issues you can go over them with the applicants before signing.I'd also recommend going through the lease, explaining any important restrictions and the penalties.

Now my concern is that its listed as a one bed one bath.

Second, it's only one thing you have to be concerned about.  

There are several concerns I have with this.1)  First, you went from ignorance is bliss to at minimum a very strong suspicion.  

Thanks Chris,I am new to this and may be using the wrong language....perhaps it's income tax that I am concerned with paying.

But I would be concerned about the realtor's lack of a sense of urgency.

There are a lot of things to consider, but if the building has been properly maintained over the years or professionally restored then there shouldn't be anything to be concerned about.  

The landlord prior to that is from Northern California (where our available rental is located), he said that there was never an issue with rent being paid, and they were great tenants.My concern is coming from employment.

Funding for the repairs is not an issue, I am more concerned about the repair process.