Being risk-adverse is a good thing, it makes you analyze a deal more carefully to ensure that it is a good one before you pull the trigger.
Yes, my recommendation would be to use your savings to make real estate (or other) investments rather than paying down your mortgage at an accelerated rate. I don't think paying off your mortgage early is a bad thing at all; it will put you on very safe footing. I just think it will stop you from ultimately making as much money as your could and eventually being on even safer financial footing.
In terms of increasing your cash reserves, I don't think that is a bad idea. I think 6 months should be fine, but if you are more comfortable having 12 months of reserves that just makes your more secure. You could also tap into those funds if you need reserves on any of your investments.
For down payments on your rentals, I think you should put down as little cash as you can. That largely depends upon what the lender wants you to put down. However, the caveat with that is that you should not leverage it to the point where it is too risky; you don't want to put yourself at risk of losing your rental. You would have to run the numbers to see where that is. Putting down less cash lets you scale up faster by being able to afford rental #2 sooner.
In terms of a 15 or 30 year loan; I think if the rental can support a 15 year loan I would do that. It would have a lower interest rate than a 30 year loan, and given that you hate debt it seems like it would work better for you to pay it off faster. Your cash flow on the deal would have to be pretty good to support a 15 year term, but if the property is paying for itself (even if you are not pulling money out) that is a pretty good result.
Of course, if you are interested more in immediate cash flow from your rentals then you should go with a 30 year loan.