What exactly do you mean by "weird" regarding the structure you have enclosed? I see nothing obviously wrong with it.
I am not sure I fully understood your second question. The interface builder allows you to match two structures by rotating them with respect to each other (i.e., by choosing different shapes and dimensions of lateral unit cells). This process is automated and produces a set of interface structures with different strain configurations corresponding to particular choices of the lateral unit cells I have mentioned in the previous sentence. This is fully geometrical procedure, and it is up to you to decide on which ones suit your needs.
In general, one usually aims at structures with no or minimal strain. In practice, one may want to avoid dealing with huge structures, and in this case one should tolerate some strain in the interface structure. But that is really up to you to decide what is good for a particular problem of your study. For Li|Li3N, I would perhaps allow some strain on the metal side, i.e., in Li, and this is what you have done. You may certainly consider an interface structure with less strain, but you will then likely need to enlarge your lateral unit cell size.
If you only do ion optimization of your interface structure, the macroscopic strain will not change for the final structure. Otherwise, you have to use in-plane lattice parameters of the final interface structure to recalculate the strain tensor.