Sometimes dogs limp or appear to be limping on the paws opposite to the one that actually hurts. If you still can't find anything, you should probably consult your vet. I doubt it's related to getting fixed, but crazier things have happened.
Legal entities have the right to own property, the right to trade, the responsibility to repay debt, and the responsibility to account for themselves to regulators, taxation authorities, and owners according to rules specified in the relevant legislation.
What matters is that the employer has the legal right to control the details of how the services are performed. If an employer-employee relationship exists (regardless of what the relationship is called), you are not an independent contractor and your earnings are generally not subject to Self-Employment Tax.