Yes it is not an illegal drug. My friend bought a suitcase full from india and it paid for his next holiday.
Yes, it will be found by the Tsa agents during security. They probably won't grab it or anything though.
However, whether you are traveling internationally, I would check with your arriving country to see whether they permit sildenafil (which is Viagra) into their country.
Yes, you can take viagra through airport security.
It is not a controlled drug. There is no offence of personal possession of it, even if it hasn't been prescribed to you.