Use Rigid Foam Underneath Sidewalk to Control Water

Your downspout extension may be installed incorrectly. It appears that the black pipe coming out of the hill is not sloped properly and is causing water to pool inside of the pipe.

Since the pipe is not draining properly, organic debris such as leaves will clog the pipe and render it inoperable. Since water takes the path of least resistance, it will back up and then come out of the bottom of your downspout instead of out the extension.

The stones are not doing very much, and they are mostly decorative. Many people put stone, or some other kind of ground cover under their decks because it prevents weeds and grass from growing.

If you want to keep the stone there, I would recommend removing the plastic tarp from underneath it and replacing it with landscaping fabric. The fabric will allow water to drain through, and prevent weeds from growing. That way, you would be able to reduce the amount of maintenance in that area.

~Is the fact that clay is rocky going to cause any problems? I have removed as many rocks as I can but it is impossible to remove them all.

The rocks should not pose an issue as long as the soil base is compacted.

Can the depth of my crushed stone base fluctuate so long as it is always at least 4" deep? After excavating and removing stones the soil bed is not a consistent depth- some areas have space for 4" of crushed stone but some areas could go as high as 8" or maybe even 12" of crushed stone. Is this ok? Does it matter if the soil bed is not perfectly even so long as it is compacted, has geotextile fabric over it, and the crushed stone is lever after installation?

The geotextile fabric is good to use as a base over the soil base. Being in New England, you will see cold winters with the possibility of frost heaving. Thicker crushed stone bases help prevent this. 4" is the minimum crushed stone you will see recommended. If possible, increase that to a minimum of 6" across the entire patio area. When laying thicker crushed stone bases, it is good to use a plate compactor with every couple inches (3" or so) of base. With 4" of base, the plate compactor can be used once it is all laid down. With 6" or more it should be used two or more times.

Also, extend the crushed stone base at least 6" outside of the area to be covered by pavers. This will ensure the pavers stay supported by the crushed stone even with expansion and contraction.

The fluctuating depth should be minimized as much as possible, but it will not ruin your patio as long as the base soil is compacted.

~My last questions are about drainage. The patio will slope down 4" away from the house over 16 feet, which I believe is a good slope?

Yes. That's fine. 1/4" vertical drop per horizontal linear foot is a commonly accepted slope for drainage.

~Lastly, the water will drain right towards my detached garage (an old 2-horse carriage house that isn't nearly as nice as that may sound), which has a concrete slab floor and no foundation. There will be about 12" between the end of the patio and the garage. My plan is to dig a canal here to collect the water and move to another part of the yard to drain. Is this a good idea or am I asking for trouble having the canal so close to the garage? The setup of my yard makes it nearly impossible to have it drain in any other direction.

A concern with the patio being so close to the garage is waterflow from the patio causing erosion and undercutting the garage's slab foundation. I would suggest installing a French drain to divert water down this path rather than an open canal. The French drain would use 4" perforated corrugated pipe with a sock buried at least 4" under the surface. The pipe should slope at least 1/4" vertical drop per 1 horizontal linear foot, the same as the patio. Cover the pipe with 4" of pea gravel and top it with larger rocks, such as river stones. The rocks will help to break up and slow the flow of water to prevent erosion of the surrounding soil. The French drain will help remove water quickly to keep the soil from becoming water logged.

If you don't have an appropriate place to locate the exit of the French drain, you can dig a dry well. It is essentially a large pit filled with gravel into which the French drain can deposit water to be absorbed by the surrounding soil.