Ah, yes sorry, I should have explained better.
A
lattice is a set of purely mathematical points in space such that every lattice point is exactly equivalent (that last part is an important constraint - you only get 14 such "Bravais" lattices). They're generated by translations ##\mathbf{R}_{UVW} = U \mathbf{a} + V \mathbf{b} + W \mathbf{c}## for integers ##U, V, W##. The vectors ##\mathbf{a}##, ##\mathbf{b}## and ##\mathbf{c}## are called primitive vectors, or basis vectors.
You get a crystal
structure by convoluting the lattice with a certain "motif". A motif is just a set of atoms which you attach to every lattice point. A motif could be as simple as a single atom, or as complicated as hundreds of atoms attached to every lattice point (complicated biological structures, for example!).
So it's important to distinguish between the lattice (purely mathematical set of points), and the crystal structure (lattice convoluted with actual stuff).
I don't think so, no. For example ##\gamma##-iron has a coordination number of 12, I think, but has an underlying FCC lattice.
Also, I'm not sure if it's best to say that the atoms are covalently bonded to each other. Perhaps someone else can advise