You can run a pond without a mechanical filter. Plants roots can do the job, you just need to add the plants and lots of them (I believe it's called a biological filter). Keep in mind I am new and haven't even filled my pond yet so I am just telling you what I have read ;) As for the fish, I would use the pond water to fill the pools to prevent shock and put some sort of air stone or something in the pools to create bubbles or the fish might die. There are many helpful people on here that I am sure will be posting more detailed and useful information soon.

zetasoul, what exactly is coming out out of the wall right now. Is it just a standard pvc stub, just the end of the pipe? Or is there a fitting on the end of the pvc? The sink tailpiece will insert into the P-trap inlet and be captured with a threaded nut. The P-trap then connects to the tailpiece, the P-trap tailpiece runs horizontally and goes into the piece of PVC sticking out of the wall. It sounds like all you have right now is just the nub end of a piece of PVC sticking out of your wall? No fitting on it? You need a trap adapter. One end of the adaptor gets glued to that nub sticking out of the wall. Your horizontal P-trap tailpiece then sleeves inside the other end of the trap adapter, a nut is tightened, and it's sealed by compression on the sleeved joint. If you look at the photo below, it the adapter is the piece in the far right of the photo. You can see the left end of the adapter is screwed to the horizontal tailpiece. The right end of the adapter will get glued to the pvc nub coming out of the wall. Here's a photo of one version of the adapter, you'll want one made out of PVC: Raehelen, just for clarification, she's using pvc, so she should use PVC glue and not ABS glue.

Hopefully you have PVC or some similar plastic pipe. The proper thing to do is to pull the vanity, open the wall, raise the pipe, reinstall the vanity. The picture you've drawn begins to look like an S trap. These are not code compliant since they can lead to emptying the trap by siphoning.

Thanks everyone for all the great input and suggestions. My reading is that the kitchen pipe is likely able to be rerouted into the wall, but the bathroom may be trickier due to the presence of the 4x4 and the shear wall. @weedmeister Thanks for clarifying that tripled counts as doubled. @ksc36 I had somehow missed that the code allows for an exception if a stud shoe is used. Good to have as a backup option. @joseph_corlett @Jake The Wonderdog Yes, the inspector has the final say. I've submitted these questions to the inspector but so have have been unable to get ahold of them. @User No 2nd floor or snow here. The bathroom wall is a shear wall for earthquake resistance (not wind), so that wall may post the most challenges. @ci_lantro Great point about the ID / OD. I'll have to double-check the measurements on-site to confirm whether this will even be possible without using a stud shoe. I will be meeting with my contractor later today and will ask about drilling through the floor. I do have a crawl space below so maybe that can be a solution. @cat_ky Thanks for the vote of confidence in cutting the cabinet. I wanted to clarify that my concern is not just about the back of the sink cabinet, but rather that I'll be making a 2" H x 2" D notch into edge between the back and the side panels, substantially weakening the supported weight. In particular, the cut into the adjacent cabinet (trash for kitchen or drawer for bathroom) will go almost 40% of the way across the back. I'm planning for a farmhouse sink + stone countertop, so there will be quite some weight. I've attached a diagram in case that helps visualize the cut. Do you still think this is not a problem? @Jeff Grenz Thanks for the context. I've reached out to my structural engineer for comment. The weird part is that I can't seem to find where on the plans they indicated the installation of a 4x4 post at that location in the bathroom wall.