@Francois Martin Happy to help!
I certainly do background checks. What I do is take all of the applications I have and go through them one by one. If someone meets all of my requirements and has great past landlord/work/person references, I then move on to the background check.
I go through TransUnion SmartMove and pay for it myself (it's around $30 I believe). I do not charge any application fee (I am pretty sure it is illegal in MA, although I see many landlords still charging). The background check is worth it because this looks at past evictions (I also have a website linked directly to my county's court systems), criminal history, and their credit score and all the bills/loans they have.
For pets, I would prefer no pets, but I certainly allow them. In fact, all of my tenants so far have had an animal. This is something I am very strict on however. I prefer dogs over cats (Cat urine is a pain to get rid of). I do not allow more than one pet and I mostly prefer medium to large sized animals. Nobody wants multiple chihuahua's running around and barking all day. In my couple years being a landlord, I have found that all of the good tenants have a pet, so I had to break my original rule of no pets and just make sure I am placing the right people in the apartments.