Yes, it is all totally possible, but why not use the trains and busses? Car rentals are very expensive! I wouldn't recommend a night in Tarifa - it is extremely windy, and there isn't much there. Extremely small town, more of a stopping point. After doing a day trip in Tangier, you could stay in a much nicer place! Like Marbella!
I paid for a day trip to Tangier and it included a bus from Torremolinos and back again. From Seville, I took a bus to Torremolinos and stayed for a month, nice town, you can find lots of places to stay, then take the tour to Tangier. You can also do a "tour" to Ronda which is a day trip. Again, you can do another day trip to Granada, or take the bus to Granada and do your own thing. Public transportation in Spain is very good, reliable, punctual and reasonably priced.
I rented a car in Torremolinos just to go to Granada which was an easy drive, but in Granada parking is very difficult and expensive. It is more of a pain to have a car there than a benefit. Ronda is also a city where parking is expensive, difficult to find, and many of the streets you can't drive on. It is so small you can walk everywhere. Really beautiful!
Be very careful of car rental requirements. A lot of people do not read all the details before renting and get stuck. You will be required to pay for insurance, no way around it. Get an international drivers license in the US and bring your US license with you. Many of the companies have a prepay gas thing that you can't get around either. Europcar is a good option. I used Centauro and had some challenges with them, but all in all it was fine. Check out reviews on the company you choose, there are a LOT of flaky car rental companies in Spain.
Another option is a car ride sharing website I used. Very big in some areas of Spain. I used it quite successfully to go from Torremolinos to Balcon De Europa (forget the town, but it was 1 hour away) blablacar!