Wiring Arduino and ULN2003 on same power supply

So, ignoring that my wiring diagram is terrible, is this alright to do without the Arduino getting cooked? Battery in the circuit would be a battery of some form (probably a 12V I've got lying around) with a DC-DC voltage regulator stepping down to 5V as both the Arduino and the stepper will be running off that.
I don't seem to be able to find a direct answer to this which means I'm probably glossing over a super basic concept as I'm in that annoying halfway house of stuff half-remembered proper education from years ago and freshly learned absolute basics.
I did find mention during Googling of noise being a potential issue and to be honest as this is for something super basic and I don't have any capacitors handy I'll deal with that if/when it causes a problem.

Yes, that is a pretty crappy drawing. I don't think anyone can advise you unless you at least add pin numbers to the drawing, and please rotate the camera so that we don't have to rotate our heads to make any sense out of it.

Edited to rotate and add pin numbers, but excuse me if being daft here but the 4 communication wires between the Arduino and the stepper driver shouldn't factor into how the two are powered?

You still have unlabeled pins so the drawing still makes no sense.

Link to the stepper, please. ULN2003 will create fumes for certain steppers.

It's a 28byj-48 5v.
5v to Vin on Arduino and stepper drive.
0v to gnd on Arduino and driver with gnds between the two linked

Nope. 5V goes to 5V pin on Arduino (the pin name is the clue there).

There are much better stepper drivers available these days, do you have to use the ancient ULN chip for some reason?

Because I have it and it'll do for what I'm doing rather than buying something?
Apologies for the confusion on the Arduino pins, misremembered what they were labelled.
Basically will powering the two loads from the same + and - be alright without breaking anything?

I guess so, provided your 12V supply and the DC-DC converter have plenty of current capacity. If not, when the motor coils actuate, the voltage could drop so much that the Arduino resets or goes crazy.

What else is the Arduino connected to? If there is nothing else connected to the Arduino that draws more than a few tens of milliamps, then you can power the Arduino directly from the 12V supply (this time using Vin). This leaves the DC-DC converter powering only the motor, so if there is a voltage drop when the motor coils actuate, the Arduino is less likely to be affected.

The DC converters are either 5A or 2A (they're tucked away in a drawer where I'm not at the moment) so that should be plenty either way.
But if I do it off the one line I can stick the converter (and everything else) in a box on the end of a wire with a switch on it which makes my life a looot easier for the intended purposes

I wouldn't recommend that at any time unless you are a very small person or it's a very large drawer.

The large currents drawn by the motor when the coils are activated could cause a large voltage drop in those long wires, causing the Arduino to reset etc. So use heavy gauge wires and some large caps (470~1000uF) near the Arduino/motor to act as reservoirs.

For a 28byj-48 5v, a TPIC6B595 is a good driver.