Much of my career as a EE, has involved batteries, charging, and fuel gauging and safety circuits for secondary batteries.
I have heard there are potential problems connecting Li-polymer type batteries in parallel -- with warnings like: "don't do this unless you know what you are doing." I would like to know what I am doing 
Here are the issues I can think of:
1) If you connect a charged battery to a discharged battery, a large current may flow from one to the other. If the batteries are initially connected when at the same state of charge and never disconnected, this should not be a problem.
Your thought process is good. Most manufacturers data sheets will publish a spec that details the maximum configuration they recommend for the number of cells in parallel. This spec takes into account variation in capacity for the cells, and how small amounts of unequal current sharing will effect cell life. In general for Li-Ion, I've seen good cell manufacturers that allow up to 4 cells in parallel. Some of the less well know manufacturers have a less stable process, and want only 2 cells in parallel, or don't recommend parallel configurations at all. So, if it is a really well known cell, Panasonic, Sony, ATB, etc. Feel free to parallel 2 even without a data sheet. Just be careful that the cells have had the same treatment for their lifetime before interconnection.
2) If the internal resistances are not equal, the charging current during the constant-current phase will not divide evenly. This may cause one cell to be charged faster than it is rated for. If you keep the charge current well below the total maximum charge rate, this should not be a problem. For instance, if I have 5 2000 mAh cells in parallel and charge with a maximum current of 1 amp, it will take many hours to charge, but with no risk over overcharging one cell.
Even if I went up to 4 amp, it is unlikely a single cell would get more than the 2 amp / 1C charge rate. If I wanted to charge at the maximum rate of 10 amps, it would be more problematic, but then I don't really want to make a 10 amp charger anyway. The constant voltage phase of the charging should always work fine.
As long as the manufacturer allows connection of parallel cells, you can easily go up to the C rate they recommend. Note that not all cells are designed for a 1C rate. A great many cells are really designed for a 0.7C charge rate. As always, Li cell chemistries are sensitive to overvoltage and undervoltage, so be sure to use protection circuitry that monitors any series connected cells, and stops charge/discharge upon overvoltage/undervoltage.
3) With multiple cells in a bundle, generated heat problems may be worse, at least if coupled with fast charge/discharge rates.
Absolutely. With typical Li-Ion and Li-Polymer cells at modest charge rates in the area of several amps, I don't expect a problem. If the cells are enclosed and can be heated by other circuitry, I suggest a safety circuit that also monitors the temperature of the cells. In general, if heating by external circuitry is unavoidable, try and make sure to mimimize any hot spots in the cells if possible.
Is there any known problem I haven't thought of? It sounds to me like if I pick identical cells, keep them connected always, and don't need to push the limit on charge discharge rates I should be fine. What about aging effects? If I do want to push the charging rate higher what do I need to do?
The manufacturers data sheet is your friend here. Many cell vendors post very abbreviated specs on their website. Usually if you sign an NDA, and can show you have a legitimate need for the information, they will share the detailed requriements with you. If you want or need higher charge rates on Li-Ion or Li-Polymer cells, I suggest you look for cells rated by the manufacterer as "Power Cells", as these are designed for high rates of charge and discharge. Of course there is always Litium Iron Phosphate - LiFePo4, which are especially good at high power charge and discharge.
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