Possible. The computers are linked between the warrant division and the drivers license agencies, but it really depends on a few factors: the county that issued the warrant, whether it is a misdemeanor or felony warrant, and whether you are applying for a license in a county different from the one that issued the warrant. I wouldn't chance it though without an attorney.
If its a misdemeanor up to year in jail and up to a $1000 fine. It depends on the county where you were charged. If its a city ordinance up to six months in jail is the maximum time. You won't be put on the sex offender list from the facts you have given. However, the prosecutor may see it differently and amend the charges.
Yes as the biological father for an adoption to go through he must terminate his parental rights. If he is not contacted and properly served the adoption could be set aside. Please proceed with caution as you don't want to adversely affect the your son.
When you are being charged with your third felony. If convicted it applies then.
Call the warrant division either at the sherriff's office where you think the warrant may have been issued or you can call the police department you think may have issued it. Either way they departments have to have the warrants because they will try to physically come to the last address where you lived to give it to you or they must try to contact you about the warrant. If you find out you have a warrant and don't want to turn yourself in you will have to hire a lawyer