I’ve recently come back from Soller and did two guided trips with Tramuntana tours (based in the port). I thought the mountain biking was ace. I’d warrant the off road route to the aforementioned Cala Deia is far superior to the road one 😀
Regarding access, my understanding is that unless specifically noted then all footpaths are good for walking on. My guide seemed to think there was little restriction on riding (unless you were on private land).
Soller or Port Soller both seem like good bases for both mountain biking and road biking. If you and up taking a mountain bike I’d recommend taking the FS.
However, if you have limited time then as others have said the road bike might be the better option. There are some very good road routes that are a doddle to navigate.