The plan was to pick a hire car in Hafia, and then travel to the Sea of Galilee/Golan region checking out (Tsfat, Rosh Pina, Banias Nature Reserve and Castle Nimrod).
Basically, yes, but you'll have to be in a hurry all the time. I personally don't love my trips in a rush.
Driving from Haifa to the Golan is about 1.5-2 hours.
The Banias reserve is a one day trip. It consists of 2 parts, the waterfall itself, and the antiques.
It is very close to Castle Nimrod, and Castle Nimrod can be covered in 2-3 hours:
The opening hours of both Nimrod and Banias are not so wide, 9:00 - 15:00 or 16:00.
About Tsfat and Rosh Pina , it depends on the activities you are looking in those cities.
As for the base, Tiberious have tons of hotels, and all over the north of Israel you can find high quality Zimmers. Zimmers are better reserved up front. Basic Zimmer with breakfast costs around 600-700 NIS (for couple), prestige zimmer can cost you around 1000 - 1200 NIS (with sauna, private pool etc). Those prices are for weekends, middle week prices are lower.
T'veria (Tiberious) would definitely be the best place to use as a "base." As for three-four days? I would think a week would be better if you want to be relaexed. That said, you could definitely do it in less time - not to worry.
As for things to do, definitely Tzfat, Rosh Hanikra, The Golan (I recommend a jeep trip in the area), and the Kineret (Sea of Gallilee). Also, you may want to consider a day/half-day trip up to Metullah and see the Hermon Mountain.
Hope that's helpful!!