Yes, so long as the probation is completed. One of the ways you are disqualified from possessing firearms is if you are a convicted felony. However, a juvenile adjudication for a felony offense is not a felony conviction. Similarly, if a juvenile charged with a felony received deferred adjudication, and the case is now dismissed and probation is completed, he should be eligible to possess a firearm. Seek the services of an attorney prior to buying a firearm, however, to ensure your particular case is good.See question
No. In order to "open carry" in Texas, you must first obtain a Licence to Carry a Handgun ("LTC"). One of their requirements for eligibility of the LTC is for the person to be at least 21 years old.
Read more here for how to obtain the LTC: https://www.dps.texas.gov/RSD/LTC/faqs/index.htm
Here is the statute for eligibility for the LTC: http://www.statutes.legis.state.tx.us/Docs/GV/htm/GV.411.htm#411.172See question