OK,lots of fun here!
The relay module is this https://www.electronicscomp.com/1-channel-5v-relay-module
OK, this one.
well, this module -- with only three terminals - is not isolated, the optocoupler is merely a novelty. It may not matter, noting my points about critical layout of the wiring. Plenty of commercial equipment, properly designed and laid out, works without such isolation.
The Nano isn't powering other stuff. I have a 9V AC to DC barrel adapter plugged into a breadboard power supply that provides a constant 5V rated at 700mAmp max.
No it does not, or at least not after a few hundred milliseconds as it heats up. It is almost useless in the same way as I explained regarding the regulator on the UNO/ Nano with almost no heatsink.
All the components including the Nano and the relay are drawing from it.
You need a regulated 5 V supply such as a common USB "phone charger" rated at 1 or preferably 2.1 Amps with a cable split out to connect the 5 V and ground to your 5 V devices.
So can I just just turn it on? There won't be any AC back current that'll go to the breadboard and fry all my components? Btw I do know about inductive loads like water pumps. I read that the starting current is 4-6X the rated current. If my pump is rated 2.4amp won't that go over the 10amp limit?
It might, but in fact there is less of a problem with a relay making a circuit with that current than there would be if it was needing to break it (arcing).
Also my pin headers are too loose for the Nano so I'm hooking it up straight with the jumper wires twisted into the Nano's holes. Could there be a sparking problem?
There will be no sparking in the 5 V circuitry, but I am not sure what "pin headers are too loose for the Nano" might mean since pin headers must always be soldered to the Nano. It is possible to assemble a final project with wires soldered to the Nano, but since you need to make multiple connections to some, you would generally either mount the Nano with its pin headers soldered into stripboard or your own PCB, or plug it into a terminal adapter such as this: