Wow so much awesome information! Thanks nILS and stryd_one! I’m not sure where to start, so I’ll jump right in:
I don’t quite see why you have all the text on both the tNames and 121_tsilk layers. Unless that’s a batchpcb thing, it’s not really a good idea
The errors you get mean that there’s something within the stop masked parts. That would be all the component outlines on tPlace. Hide the layer and run the DRC again, and you’ll see.
They actually require layer 29/30 (tStop/bStop) as well as the silk layers (they are created with a script that recommend running). The errors are actually for layer 29, and, you’re right, if I hide them, there are no errors. Since their DRC step passed on their end, I assume it’s probably nothing to worry about?
You’ve done some things in the layout that I cannot quite see a reason for, which isn’t necessarily bad though 
Whoops
Can’t say I’m too surprised since I’m a fairly big PCB CAD newb 
Footprints of 100nF/330nF caps tend to be 200mil - you might wanna check the parts you plan to use.
I sort of guestimated based on the caps I purchased for the protoboard design of the board. They are the ones that stand up and are usually round at the top. Sort of like a stick figure without arms. Radial I think? The symbols you have are for axial right? Or no?
I made some changes to the layout (the caps are supposed to be flat as in parallel to the PCB) kinda answering one your questions: Yes, you can route the signals like you asked. Why wouldn’t you? As long as the parts are on the correct net you can basically wire them however you like (this is a half-truth. While theoretically you can, you wouldn’t want to do it for various reasons. A bypass cap that is supposed to smooth things out for an IC don’t help much on the other side of the board, HF traces (and all others really) should be as short as possible, well you get the idea). Before you start actually laying down traces, move the components around press F6 and see how the airwires change. That’s another way of looking at the answer to your question ;D
I think I see what you are doing. So basically, it doesn’t matter if I connect the +5V with the -9V right at the big capacitors? I didn’t think so, but didn’t want to waste a board design trying to find out. So thanks for clearing that up, though I was wondering if you can elaborate on what you mean by parallel and what lengths should be short as possible? Do you mean all lengths or, in this case, the small caps? If all lengths, now that I can connect the wires to the big caps, I can shorten the circuit even more. Actually, your board design is most of the way there it looks like 
F6, by the way, didn’t work. That turns the grid off and on for me, but I assume you are referring to the RatsNest feature? I used that on a regular basis, but in this case, I wasn’t sure if I could trust how it connected the wires since I wasn’t sure about when I could combine voltages as noted above.
Finally, I noticed that you used a ground plane. I had considered doing that (just like with the Bankstick board I’m working on) but I was a bit confused about how that would work or if I needed it for a power board. I probably should’ve thought about it more, since your solution looks obviously exactly like it should look
I assume the general rule of thumb is to ALWAYS use a ground plane? And probably almost always use an isolation plane for a two layer board?