Yes, the generalized coordinates $(q^1,\ldots, q^N)$ are assumed to be independent, i.e. no constraints. (This is a common assumption in textbooks.) They constitute an arbitrary local coordinate system for the configuration manifold $M$; in particular they may not be orthogonal.

See also e.g. this related Phys.SE post.

Yes, the generalized coordinates $(q^1,\ldots, q^N)$ are assumed to be independent, i.e. no constraints, and the cotangent vectors $(\mathrm{d}q^1_p,\ldots,\mathrm{d}q^N_p)$ at each point $p$ are linearly independent. (This is a common assumption in textbooks.) The generalized coordinates $(q^1,\ldots, q^N)$ constitute an arbitrary local coordinate system for the configuration manifold $M$; in particular they may not be orthogonal.

See also e.g. this related Phys.SE post.