Simple circuit with operational amplifier (intended to be ECG) not picking up actual readings, only noise
I'm almost afraid to tackle this one, but here goes.

Stop. Just stop. Forget EKGs. Get a cheap DMM, find a web page that deals with really simple op amp circuits, and start building really basic stuff, like a voltage follower, then a non-inverting amp with a gain of 2, and work up from there. The LM358 data sheet has some very basic circuits that you can try. At this point there is simply no way help you "fix" what you've done. Your schematic makes no sense at all, and the Fritzing does not correspond to the schematic.

In the schematic, there is no pin connected appropriately. Not one. First, you show an OP37, then you say you don't actually have an OP37, you have an LM358. Why did you do this? You have +5 going to the V- input, and D9 going to the V+. The LM358 doesn't have trim inputs, and on the OP37 that's not how to connect them. And the output doesn't go anywhere. Finally, an op amp is not run "open-loop" - that is, without feedback from the output to at least one of the inputs. So just running your signals directly into the inputs is not going to work. Like I say, there is not a single pin connected appropriately.

The Fritzing shows, for instance, +5 connected to pins 4,5,7 and back to DIN. There is no ground connection to the breadboard, so there is no ground connection to the op amp.

I'm sorry, but when you're this far off, there's no easy place to start.

Cease with the EKG for now. Concentrate on making a few simple circuits that work, and only then move on to something more complicated.