The best source of information on starting a business in Minnesota that I'm aware of is this:
 http://www.ci.moorhead.mn.us/eda/pdf/A_Guide_To_STARTING_A_BUSINESS_IN_MINNESOTA.pdf

I believe you will find that if you do business as a sole proprietor in your own name, there's really nothing you have to do.  If you do business under a name other than your own, you have to file a Certificate of Assumed Name with the Minnesota Secretary of State's office.

You do have to pay income and self-employment taxes on the money you make (that's true anyway), but you get to deduct legitimate business expenses.

I'd take the time to get bonded and insured as most people won't hire a complete stranger without some proof of legitimacy.

I would definitely utilize a service like this at times, even just for a day or two.  However, I would definitely want references and I do most of my research online so a website with credentials, etc. would be good.

I used to travel for work more than I do now and utilized someone who offered services like your are offering.  I don't need you anymore, but would also advise along with getting bonded, etc. for business get some kind of vet/grooming training or having had experience working in one.  The person I had was fabulous with my dog, took it home for the evening to play with her dog and family, took him to the lake, etc. Also took wonderful care of my house providing extra services. You have no idea how wonderful it is coming in after a hard week on the road and a long day of crappy air travel and finding your mail stacked, carpet vacuumed, and your dog happy to see you. References would be a must. Rememebr this is a job, and you are in someone else's space.  Act accordingly.