RE: US 20060112899 A1 - I'm developing a similar product, not specifically for pet care use. Would this be viewed as a totally separate and legally patentable product, since it would be more all-purpose?
The patent document you referenced is not a patent. It is a publication of an application for a patent. If you look it up in the USPTO public PAIR database you will see that after its initial rejection (almost all applications get an initial rejection) the applicant made no response in the required 6 month period. Therefore the application went abandoned. The examiner cited US 5639532 against it.
More generally (but imprecisely), a patent might cover a thing, a method of making the thing or a method of using the thing. If someone patented a type of wipe (a thing) a different use for the wipe still requires a patented wipe. The new use might be separately patentable but to carry out the new use one would have to get a patented wipe. Addressing another aspect of your question - what makes it have a wider use? If it just that you explain how to use it more broadly, you have not invented a new thing. If making it more broadly useful means making it structurally different, then it is the structural difference that might make your design patentable.