Wiring – Can PoE cause interference with neighboring non-PoE Ethernet cables

TL;DR it should be OK to run them together.

The signals from a security sensor are typically very low frequency because they're generated by the switch within the sensor opening and closing as the sensor activates. So for a motion detector, it might open and close a few times as you cross the field of view; a door or window sensor would switch as often as you open or close its corresponding door/window, etc.

The signals from a remote keypad are higher frequency because they have to communicate more information, e.g. the text to display on the screen, or which buttons were pressed on the keyboard, but they're still going to be low frequency¹.

Twisted pair cables, such as Cat 6, are fairly immune to outside electromagnetic interference, and coax is even better thanks to its outer shield. And since the signals they carry (e.g. Gigabit Ethernet, TV signals, etc.) are very high frequency, the chances of interference with the security system's signals are very low.

^{1. The value 300 bits/second is emerging from my long-term memory, but I could be conflating it with the speed of the soft modems on security panels}

Darn near none. Even less if you do it right and run conduit so you can replace it when you can't live without Cat9, or 12 pair fiber, or whatever everyone will have in 5 years.

The whole point of the fancy-pants wire is that it is, in fact, resistant to interference pickup, in places a lot more challenging than your house. Though I hardly see the point of Cat6 - it can barely run 10Gbit, and Cat5e runs gigabit just fine (I have over half a mile of it (in sections less than 100meters, of course) connected to switches that report error rates, which are pretty much 0)