The state of TN does not require that trailers be registered if it is being used for personal use. If it is used to make money then it needs to be registered for commercial use, as does your vehicle. Although that might not be enforced, it would still be a good idea to do. You never know when someone is going run a red light and broad side you when you have all of your equipment, and you get a citation for improper vehicle registration. Its like insurance; you hate to pay it but you've got to have it: Cost of doing business.
The tag prices I listed earlier where for Rutherford county, so they probably won't be the same where you are.
As for Taxes: TN does not have a state income tax so its just federal taxes you have to deal with. Since you are operating as a sole prop. the business will not have to file a tax return. all money made it considered your personal income. I would highly recommend that you talk to a CPA about your tax questions. If you can find a good one they are worth their weight in gold.