4 days in and around Hakodate

I have 5 days/4 nights to spend in Hakodate from May 9 to 13.
Just wondering if should be staying all 4 nights at Hakodate?
Is it more worthwhile to hire a car for 2 people or use public transportation?

This is what i have in mind roughly:
day 1: flight into hakodate arrives at 10:45am. Drop luggage at hotel, go to goryokaku park, go to motomachi, mount hakodate for night view
day 2: Matsumae (should i take train or hire a car?)
day 3: Lake toya and noribetsu (should i take train or hire car? stay overnight?)
day 4: around hakodate : visit morning market, bay area, what else to do?
day 5: noon flight out of hakodate

where else is reachable for a day/overnight trip from hakodate besides toya and nororibetsu?

day2, car offers you more flexibility and take 2hrs one way while train+bus will take about 3.5hrs, less flexibility and 2090yen per person.

day3, you might be able to do both with car depending on your pace. But I would recommend taking train and staying overnight to enjoy the onsen.

day4, you could try onuma park if you are interested
http://www.japan-guide.com/e/e5356.html . If not, you could cut one day and fly back.

After having been to Hakodate several times, I decided to dedicate 3 days by rental car just to explore the environs of Matsumae and Kameda Peninsulas. From Hakodate, I circled the coastline counterclockwise up to Mori, cut west to Esashi, then returned along the coast to Hakodate via Matsumae.

Couple of interesting notes:

1. Little known fact is there is a monkey onsen in Yunokawa:

http://www.hakodate.travel/en/things-to-do/top7/yunokawa/

However it's at a flower park, not a natural environment like Jigokudani.

2. Mt. Esan is an active volcano with a walking trail to a hot spring area. The steam pits reminded me a lot of Unzen or Kawayu area in Hokkaido.

3. Onuma area has a lake and hiking trails. The JNTO .PDF on Hakodate has more details on Yunokawa and Onuma:

http://www.jnto.go.jp/eng/location/rtg/pdf/pg-102.pdf

4. The area outside Hakodate is famous for wakame and large blue surf clams. You can see wakame drying on gravel beds along the coastal towns, or buy surf clam sushi at supermarkets. There is also a big wakame museum outside Hakodate.

5. Esashi town is famous for mackeral on hot soba noodles. Most of the best preserved mackeral for this dish in Japan comes from here. One notable attraction here is a museum inside a full size Ironsides-type ship replete with cannons.

6. Matsumae is famous for hokke fish. Unfortunately, the town is so small I could not find any restaurants which served it cooked! This town is best seen during cherry blossom season which peaks mid May due to the many varieties of sakura you can see here.

Now this trip is not the first thing you should do on a visit to Hokkaido or Hakodate. I combined it as part of my main goal to drive around the entire coastline of Hokkaido.

Arigatou Gozaimasu Guear-san and Guesto-san. I will consider your suggestions :)