I am in almost the exact same position as you and have wondered the same thing. I think you should in general not try to time the market. You should lock the rate because you are planning on buying and the locked rate will help guide you towards a house that you can afford. The rate already takes into account future moves by the fed. The only reason that you are interested in the probability of the rate change and how much is so that you can understand what type of risk you are assuming by waiting to lock. You can use futures to help quantify this understanding. If I were to market time, though, I would guess that the likely upcoming rate drop will cause mortgage rates to increase as it did last time due to fear of inflation.
I would also suggest not putting down more than 20% if you have the tolerance. Mortgage debt is considered good debt due to the tax breaks that you get. Those tax breaks could disappear some day but in the mean time you can take advantage of them. At that interest rate I'm guessing you are looking for a 30-year loan. You can think of a house purchase as an investment in a mortgage-backed bond. You can earn 5.75% on your money by paying down the mortgage faster or you can invest it elsewhere in a well diversified portfolio according to your risk tolerance to earn 6 - 11%. Keep in mind that you can deduct mortgage interest against regular income while your investments will largely be taxed as long term capital gains, making the reward/risk very attractive. Investing the money only makes sense if your time horizon is long and you shouldn't be buying a house if that's not the case. I would say that your time horizon needs to be at least five years and historically speaking ten years actually has lower risk than paying down your mortgage due to inflation. Futures are actually currently predicting that inflation is at least as big of a risk to home values as housing price drops. If you are going to go this route, then be sure to include some international assets in your portfolio to hedge against inflation. Any less than 20% down, though, and the benefits start to disappear (no tax deduction on PMI, higher loan cost, etc).
That being said, I'm not a financial adviser and even if I were I'm not your adviser. The point that I'm trying to make is that before you decide how to purchase your house, you should either do your own research or hire a financial adviser to help you.