Generally speaking, that is a sound method; however, you will need to consider the issue of repeated households. I suggest reading this discussion for more information on pooling CPS data. First, I recommend running your analysis with and without repeated households (e.g. limiting to MISH=1). Then, you can compare your results to see if the repeated households are biasing your estimates.
As for the correct weight to use, you will still want to use WTFINL for non-supplement data and WTSUPP for supplement data (e.g. ASEC samples).
The different population estimates do matter within years. Since you are calculating percentages per year, you should sum the weighted counts across months to get a weighted annual percentage. Based on your example, a month in 2012 with 50 doctors out of 100 total people should receive less weight in the 2012 annual percentage of doctors than a month with 100 doctors out of 200 total people. However, weighted counts do not matter across years (e.g. 2013 vs. 2012), since you are comparing percentages.
Hope this helps.