So in LR 1.x one could select a filter and you could see on the keyword list or on your folder list a readout of how many images in either the particular keyword or folder matched that filter setting. Will LR2 do the same thing?
Yes. Just add another column in the filter bar, or change one of the existing columns to a keyword filter.
Now, you will get a list of keywords in what ever folder folder or collection you select along the left and as you select other filters, say, date, the keyword list is narrowed down again. Like all the filters, count is provided.
Also, when using the filter bar to find images in LR2 every time I select a different folder or keyword the filter bar defaults back to No Filter which I personally find HIGHLY annoying. Again, any way to change this?
Yes, it does, but some may consider that a feature. Lightroom remembers the filter setting for the folder or collection. If you switch back to the previous folder, you will see the filter settings that you had pop right back ("sticky" as madmanchan mentions), so it is not defaulting to 'no filter', but rather to the filter you last set for that particular folder. As mentioned above, you might want to try adding keywords to your filter bar.