Patching small-medium drywall hole with wall cutout
You are on the right track.


Do I need to use drywall tape, or could I get away with just using the drywall mud? Use tape. It bridges gaps and reduces cracking.
Will I need the standard 3 coats of mud? And the wider knives? (I only have a 2" now.) Maybe. Try 2. See how it looks. I would get at least a 4" knife though.
How do I avoid bumps from the patch (i.e. tape &amp; drywall mud raised on the wall)? Care. Slightly bevel the edges of the insert and the existing wallboard before tape and mud. Sand between coats. If there is a lump, remove it before you let it set. Mostly, practice.
What would happen if I just used the furring strips, and filled the gaps with spackle? Don't. The texture will never be the same. And a spackled hole that size will shrink and expand at a different rate than the surrounding wallboard, leading to cracks. Your current approach is a good one.
Anything else I should be asking? Sounds like you are on track. Patience is the key. Let the surface build slowly and dry in between. Damp sanding of mud is often most effective. Don't forget to prime the new mud.