If you're walking with a tent and pitching it up somewhere high in the mountains (this may not be legal, but if you're well behaved nothing will happen), you won't have a problem with crowds. Very few people hike with a tent. In general, as soon as you're more than ~2 hours from the nearest car park, you're out of the crowds. That being said, it may be difficult (and illegal) to camp in the wild, and you might depend on camp sites. With a small backpackers tent, you won't have a problem finding a spot, but you will be sharing a camp site with car tourists with big tents, electrical light, music, etc; those camp sites are in civilisation.
Bears are rare in the Spanish Pyrenees and non-existent in the French ones.
You're very lucky if you see a bear.
There are no pumas. Wildlife is not a hazard for hiking in the Pyrenees.
On the French side, by far the best maps are the 1:25.000 map series from French IGN. However, for a long trip you might need many sheets to cover the route, so you could choose the 1:50.000 map series instead. On trails, these are usually good enough. The French maps cover part of Spain, too.
On the Spanish side, maps are less reliable. My experience with the official maps from Spanish IGN is that many hiking trails are simply missing on the map. Special park maps, only available for touristic regions, may be better. I haven't been to the Pyrenees, and can't recommend specifically.
Where to hike
I haven't been to the Spanish side of the Pyrenees. From what I've understood, they're quite dry. A friend who has lived in Toulouse (close to the French Pyrenees) recommends Grande Randonéé 10, which runs close to the border on the French side. The entire GR will be too long, but you can do a segment of it. GR's are well-marked with red-white markings.