1) What thickness is best - Home Depot had some 3/4 I think but it was really heavy
3/4" is heavy but I've always found 1/2" to be too flimsy. 5/8" is available if you look around but does not save that much weight and often is more expensive.
All things considered, I think 3/4" is a good choice.
2) Where do you buy it - Home Depot, Lowes, cabinet makers?
I buy it from Home Depot or Menards. Cabinet makers likely carry higher quality but the differences are probably not significant for keeping herps. I only use melamine in lidless rack systems or in cages with little moisture so the quality does not matter. If I needed extra moisture resistance I would not consider melamine in the first place.
3) How do you physically attach the sides - screws and glue?
Screws (pre-drilled and counter-sunk) and glue or my bisquit joiner. I rarely use dado or rabit joints.
4) What do you do with the unfinished sides?
Iron-on melamine edge tape.
5) Sliding glass front doors OK or do you need some kind of bracing for wide cages (looking at 5 to 6 feet width for my next two cages) to prevent sagging and binding of the glass
If multiple cages will be stacked 5 to 6 feet wide may be too much for sliding glass. However, I like sliding glass so the center can be braced with scrap product.
A few more comments:
What did you not like aesthetically about your plywood cage? I actually find it easier to work with a make an attractive cage with than melamine. Maybe some of us on this forum can make your current or next plywood cage more professional looking?
What species is this for? I really don't like melamine for high-humidity conditions.
Lastly, I often used the bull-nosed melamine shelving from Menards for most of my melamine projects. It limits the width to just over 23" so that my not work for you. Once you buy two 8' by 23.5" pieces it's cheaper than a full sheet of melamine.