Would using ''K9'' as part of my company name conflict under USPTO if another company had the same name, but used "canine" instead of "K9"?

Quite possibly. The key test for trademark infringement is whether an appreciable number of ordinary consumers would be confused into believing that your product came from the other company. Since the names sound identical and since K9 is a customary abbreviation for canine, that answer might be "yes." The answer will depend on more facts and also likely on surveys of consumers. Ultimately, you are exposed if you adopt the similar name and may be subject to suit. You could win that suit, but you could be forced to spend hundreds of thousands of dollars in legal and expert fees to defend yourself.