You can use the computer algebra package "Singular" to create a bunch of examples of your own. The resolution of ADE-singularities (the easiest examples) in dimension 3 are essentially the same as in dimension 2 and are probably not treated for that reason. To work out the resolution of an A_k-singularity (x^2+y^2+z^2+w^(k+1)) is a nice exercise. Several people tried to convince me that starting from dimension 3 in many applications (e.g. calculation of the hodge structure on the cohomology) it might be easier to work with the singular variety itself rather than pass to the resolution.

In the above paper the Picard lattice of a very general member of the family is computed. Although this is highly non-trivial, it is definitely easier then calculating the Picard lattice of a specific member in the family. (The OP asked for specific examples.) I.e., for a very general surface in $\mathbb{P}^3$ of degree at least 4 one has $Pic(X)=NS(X)=\mathbb{Z}$ (theorem of Noether and Lefschetz); for a concrete surface in $\mathbb{P}^3$ the calculation of $Pic(X)$ is more complicated.

In general $omega_X$ is not the restriction of a line bundle on $\mathbb{P}^n$. An example is the twisted cubic curve in $\mathbb{P}^3$. In this case the canonical bundle has degree -2, whereas every line bundle obtained by restriction has degree divisible by 3. You can have a look on the final section of chapter IV of Hartshorne. He gives a discussion which pairs (d(C),g(C)) are possible for smooth space curves C. In particular, it is shown that for fixed d there are many possibilities for the genus of C. (Provided that d is not 1 or 2.)

Maybe you could give a hint what you know about K, (since you claim you do not have a presentation in terms of generators and relations).