Being convicted of felony will preclude you from becoming an attorney in the State of Texas, depending on type of felony and whether you were placed on probation or deferred adjudication. I need more information about the type of case and actual terms of the conviction before I can give you a definitive answer.
However, if you are already a licensed attorney, a felony conviction can bring a range of consequences to bear from a public or private admonishment all the way up to disbarrment, depending on the type of felony charged. Again, I need more information about the actual charge and imposition of sentence before I can give a definitive answer.
Best solution is to call your local State Bar Hotline and ask them for advice. They can direct you to the exact person you need to speak with to get the answer for your state.
There is insufficient information in your questions to provide fully informed and reasoned responses. Generally speaking, it would depend on the bar admissions requirements for the particular state that you intend to practice. Some states have a more vague requirement such as the applicant for admission must have "good moral character". Other states, conviction of a felony is a per se indication that you do not possess the relevant character trait. It is possible that an applicant can still be approved if the felony was an isolated event, occurred many years prior to one's application for admission, or involved mitigating circumstances.
Your second question is equally difficult to respond to because it does not identify sufficient information concerning the crime, the circumstances, or the jurisdiction. Again, generally speaking, the disciplinary sanction imposed by a particular authority can range from reprimand to disbarment. There are too many factors within each jurisdiction that would control the analysis and too many jurisdictions to summarize. Usually, conviction of a felony by a practicing lawyer will lead to suspension at a minimum through disbarment in more serious cases.