Well.... I guess it's up to you as to whether or not you want your boundaries topologically correct. I didn't realize you had so many errors, but the only way to get experience in something is to give a try. Just be sure you back everything up before hand. (I assume you did that before experimenting with integrate, right?)
At any rate, you can build a set of topology rules such that your polygons not have gaps or overlaps; often times if you set your tolerances right, the validation process takes care of a lot of errors. Start here and good luck
That should just about do it....