First, someone else mentioned east of 82nd. That's where I live, right on the edge of "felony flats". We see the occasional car abandoned on the street and did have some garden tools that we left out stolen. otherwise, no issues. The crime rate is highly overstated, and the name is a holdover from olden days. I am reliably informed that Lake Oswego (the expensive area where the pro sports players live) has a much higher crime rate than our area. Its just that the prostitutes there have nice places and everyone turns a blind eye, rather than the ones on the street on 82nd which get cracked down on. Portland really doesn't have a "dangerous" area. It's more homogenous than anywhere else I have been. I have never once been threatened by anyone here in PDX, and have never felt unsafe in any part of the city at any time of day or night. Except during Occupy events when the police were behaving with unjustified aggression. But I dont count that.
If you want to look for information on a specific area that you are thinking of moving to here, try Portlandmaps.com. Very handy tool.
I would recommend that you attempt to stay within Multnomah County. The suggestion someone else made of Oregon City, I would not recommend. Oregon City is in Clackamas county, which has just been voting to refuse taxes and decrease funding for schools and services. Anyone making use of public assistance of any sort will have a harder time there, in my experience. I think for a cheaper option, you might look to Gresham, outer East Portland. Rents are almost always cheaper out there. You might also look at Vancouver Washington, which is kinda part of our greater metro, but also seems to be cheaper.
The cats are going to make things a challenge. I have a family member looking for housing, with 4 cats. She is finding a lot of places will take 3, but 4 seems to be a deal breaker. Or maybe its 3, and they only allow 2, I cant remember. I guess arbitrary lines have to be drawn somewhere.