I am going to be neutral when answering...
In Basque country, does the youngest generation commonly learn their own language as their native language - or do they simply manage to learn it later?
Well, the sociocultural trend in Europe is to go back to smaller entities. I won't make a whole lecture about it, but many people find it comfortable to look for their roots, try to protect themselves in way the superglobal culture cannot reach them. That is a wonderful topic, but let's go back to the main issue. Younger generations in the Basque countries are willing to be fluent in both languages, and reach easily fluency in French and Spanish. For Basque, a real effort is done, started by the previous generation which started to dig up stuff to revive Euskara (songs, literature, bilingual stuff, etc.). The previous generation missed the fact that they were not taught Basque either at home or at school, and felt a real cultural loss. Today's generation is really eager to preserve Euskara, as the number of bilingual speakers (in %) is increasing.
In Basque country, does the youngest generation commonly learn their own language as their native language - or do they simply manage to learn it later?
Yes, the Basque government (as well as, one might not know because everybody's keen on criticizing, the French government) finances ikastolak, which are private school where the courses are run only in Euskara. What is interesting is that even if your parents are not euskaldunak (Basque speaker), you can go to these schools, and their numbers are increasing.
On the French side, Basque has entered curriculum as a language that can be taken at exames like the baccalauréat.
Is the Basque language a source of pride for the Basque people or has it been otherwise...?
That's a really interesting question. I may have sounded really optimistic in my first two answers. But if you go to the Basque country (even in the Spanish side) and compare it to Catalunya, you'll see a huge difference. Catalan speak more Catalan than Basque Euskara. Maybe because Euskara is really different and takes time to be learnt, because it is a really different culture, I don't know.
There has also been a problem of guilt/shame during Franco, when kids were canned if they were caught speaking Euskara at school, when people were not hired if they did not master only Spanish, etc... Basques lost touch of their languages, feeling it was more of a burden for them. Now times have changed, and in the newly open world, where uniformity tends to be the motto, I think people have to dig up the last inches of individuality they can find, and try to connect to communities, smaller networks, etc... The second revival of Euskara is certainly the result of this phenomenon. I may also suspect this tinge of stubbornness Basque always had not to disappear in the process of history
Zoroa