I have painted more than a hundred miniatures from various armies using Army Painter Strong tone in the past two years, so maybe I can share some insight.
1. After you dip them, how thin is the covering left on it? Is it able to be painted over or do you need to hit it with the Matte Varnish before trying to highlight?
It doesn't take paint very well. But then again, I rarely paint anything after dipping, with the exception of details that are easily obscured by dip such as eyes of human-sized models. The dip is there not just to add shading, but to make any colour changes smoother. Add some highlights after dipping and suddenly the mini isn't as smooth anymore. I think it's a better idea to paint the highlights before dipping.
2. Looking at using the Strong Shade color over the Rat Fur Brown, will it shade enough to where you probably don't need to dry brush highlights on top of fur or does it still need it?
I am not familiar with the colour, but as a general rule, if I'm not sure the colour light enough, I make it a bit more lighter. Also known as "when in doubt, add Bleached bone" rule of dipping. Either way, the strength of dip depends on the surface of model. If the fur texture is properly defined, the dip will sit better and become stronger in crevices.
3. Is it really necessary to use the Army Painter Matte Varnish sealer spray or can you use other brands? I've read that it doesn't react well to other company's products.
Although I have heard a lot of good from Army Painter Matte Varnish sealer, other products work as well. Personally, I use brush-on Vallejo Matt Varnish and it works fine.
4. What colors does it do well on / not do well on? I'm looking at a Clan Pestilen themed army, but don't want one that is extremely muted and dark (as they came out with the Miniwax dip).
Dip works best with light browns and bone colours - and depending on the look you are after, several greens, yellows and reds. Colourwise Clan Pestilens sounds like a perfect candidate for dipping, similar to pretty much any Nurgle army. For my Nurgle Daemons, I chose a handful of Foundation paints to build my basic colours from. Khemri brown, Gretchin green, Tallarn flesh, Iyanden darksun and Dheneb stone are all quite earthy and colourless tones, hence perfectly suitable for dipping. While they are all somewhat light colours, I decided to mix each of them roughly 1:1 with Bleached bone, to make them even lighter (the lighter the colours are, the more you benefit from the shading). Obviously every model was undercoated white - and thanks to Foundation paints, they were very easy to paint on top of white.
Finally, my Golden rules of dipping:
1. Always use a brush to apply the dip
2. Always apply a coat of matt varnish afterwards
3. When in doubt, add Bleached bone
EDIT: Unless you have noticed, Army Painter website has a new
Skaven Clan Pestilens gallery added recently. While I do not agree with all their colour choices, it is a good example of what I mentioned above: painting the models in much lighter colours than you would normally do.