I will quote John Carmack here:
You will eventually regret any use of Euler angles.
In my experience, I have found this to be true. Euler angles are terrible for:
- Relative orientation - parents to children, children to parents
- Camera's - it's good to use a matrix, better to use a quaternion
- Animation - matrices will have to be transformed to quaternions to be slerped anyway
However, if you want to store transformations as user-changeable text it's best to use Euler angles. Quaternions are difficult to wrap your brain around (4D space? What!) and a matrix looks like a mess of numbers to the average user.
But even here, you will probably regret their use eventually.
To answer your questions:
Quaternion camera with Vec3 position. However, if it's quick and dirty, I'd use a matrix camera with a lookat from a normal to the target.
FPS-style mouse look
Quaternion camera with Vec3 position. Mostly because a quaternion camera doesn't have gimbal lock.
some kinematic routine
Quaternions for rotation, Vec3 for position. Quaternions can be lerped (fast) and slerped (quality). The only downside is that they will have to be converted to matrices before being uploaded to a shader, but the benefits outweigh the cost in this case.