Struggling with Symmetry

Symmetry is mainly about training your eyes to recognize it, but there are several little tricks that can help. I'm no master, myself, but just like you I have come a long way since I started and I have had a lot of use for these:
*Build a good core, fix placeholders for the eyes and consider to fix some sort of stand, holder or handle to the core. This will ensure that you don't accidentaly deform the entire head by holding it while working and will give your eyes a known line to lean on.
*Get the profile down as soon as possible. When you have a good center to work from it's easier to get the sides even.
*Keep checking your work from different angles, turn it upside down, look at it in a mirror etc. This makes it easier to discover flaws early and it leads into the next trick.
*Distance yourself from your work. Try to forget that you are working on a face and think of it more like a dead object. The mirror trick can help, but to take pictures and look at it on a screen can be even better. Taking breaks and then come back with fresh eyes can help too.

For the second question, that will depend a lot on what materials you are using, what tools you have available and what you used as a core. If you have plenty of space, simply adding a smooth layer of clay is a good start. If you made your core too small and the hole need a lot of widening, some sort of power tool may be the best way to go.
For final finishing, the same methods that you use for the outside will work, use the same primer and the same sandpaper to get a good finish.