Can I open my own business without impacting my H-1b status?

To work in your own business, you need to have an employer-employee relationship with the company in such a way that you can be terminated. So, unless there is an independent board of directors that you do not control, it is unlikely that you will be allowed to work in your own firm.

The terms and conditions of your H-1B employment are limited to your H-1B sponsor; you cannot work for anyone else without USCIS authorization. In addition, H-1B employment requires an employer-employee relationship where the employer has the right to control the work of the employee and the right to hire/fire the employee. If your clinic is a corporation where you are not the sole owner (but you could be the sole employee), it might be easier for the corporation to show an employer-employee relationship, and thereby sponsor you for an H-1B. You could consider two part-time H-1Bs, one with your current employer and one with your own clinic, under the conditions listed above.

Your status is tied to the H-1B employer. Working for yourself in a new business would probably not work without, at the very least, a request to "transfer" the H-1B to the new employer, which is you.

You can apply for a concurrent H-1B visa sponsored by your own dental office while keeping your current H-1B visa.

An H-1B worker is only allowed to work or engage in labor for the company that sponsored them for the visa. If eligible, another H-1B petition might be needed to allow an H-1B worker to start new employment (or to even seek part-time employment). But be wary that the USCIS eliminated the ability for self-petitioning H-1Bs in 2010. There may be alternative structures for an H-1B worker who has some ownership interest, but you will need to speak to an immigration law specialist to understand how.

No, you would need to have an H-1B to work there, either a new one or concurrent with your existing employer.

No, as an H-1B employee, you are authorized to work only for your employer. The H-1B requires an employee-employer relationship; this relationship is generally disproved through ownership. There are some instances in which it is possible to structure an entity and still qualify for H-1B, but this is challenging and not what H-1B is meant to provide. If you are from an E-2 treaty country, E2 would be a great visa for you.