As Gas has said, you would need to register a trading name for your business with the Companies Registration Office if the name you use is anything other than your own. As a sole trader, this would be your sole obligation to the CRO. You would not have to file accounts etc, as you would if you were a registered company.
AFAIK, you can still claim Jobseekers as long as you let them know of any days you have been working. The allowance normally payable for these days will be deducted from your payment.
It would also be important to check the position regarding your PRSI contributions (or stamps as they used to be known). It used to be the case that you had to sign on for a FULL week to avail of a PRSI credit (which counts towards your contributory pension entitlement and any future welfare claims). By effectively signing off for one or more days a week, you may be losing a pension credit in respecet of that week. While this wouldn't matter in the short term (or if you're paying PRSI as a sole trader), it might have an impact if you were doing it for a prolonged period.
As with any working person, you are required to make an annual declaration to the Revenue of all income received in the tax year. This obligation doesn't change if you are part employed/part in receipt of welfare.