If somebody is not the Dragonborn, would they be able to learn the dragon language and use shouts? Lore-wise?

Ulfric did.

Yes. The Greybeards are not Dragonborn, yet they know the ways of the voice. Ulfric Stormcloak learned about it from them as well. If you're role playing a character who is not Dragonborn but can shout, he/she would have to have a history with the Greybeards. Though, I am sure there are other means to learn it, but I am unaware of any other ways to learn the ways of the voice.

Yeah. It just takes years of study for them to master a shout.

It takes many years for non-Dovahkiin(damn... I've been spelling that word so wrong for too long...) to learn a single shout, but they can learn shouts. Just takes years or even a lifetime if your an orc. This is assuming all races can learn shouts. Orcs usually live to be 60.

All Nords can learn how to shout, but if other races have the ability as well is unknown to us.

It took Ulfric ten years to learn two full shouts. Note that he started his training as a child, and therefore probably learned it faster than an advlt would.

The Greybeards know more shouts, but they are very old, and I actually think they are way older than ordinary humans. If you look around High Hrotgar, you'll find an awful lot of regeneration potions lying around.

Every race has the ability to shout and learn shouts, it's just that as far as Tamriel is concerned shouts are only known to Nords and only a part of the Nord culture. So that's pretty much why only Nords learn to shout. We don't know how old the greybeards are or how old they can become. Somebody like Tiber Septim lived until 108 which is old for a human sure, but not impossible so there is nothing to really suggest that the greybeards will live forever or even impossible long times.