if I need to add an LED strip I would need to connect it in parallel to the motors, right?
Also, some people mentioned having to connect a capacitor as well but I'm not sure why.
That's for smart LEDs which are driven by a CPU to make colorful animations, not for lighting LEDs.
Will I need any sort of short circuit protection when dealing with LED strips?
Adding a fuse is never a bad idea...
Will the lights simply be dimmer (which is totally fine for my needs) or will they simply not work at all?
All 12V LED strips I've come across are built from series strings of 3 LEDs with a resistor. You can infer this from the documentation before buying when it says "strip can be cut every 3 LEDs".
Since Red LEDs will light up with about 1.6-1.8V, three of these in series means we add their voltages so three times 1.8V still makes less than 6V and you can light the strip with your 7.2V 2S LiPo. It will be less bright than when powered from 12V, but it will light. It should also work with orange and yellow LEDs which have voltages below 2.2V, green would be pushing it since it has a bit higher voltage, and white/blue won't work since these require 3V per LED at least, and you don't have 9V.
If you have a red/orange/yellow strip that lights on 7.2V, you can change the resistors to make it brighter. Remember I=V/R. If your red strip has 330R resistors you can put 100-120R instead. But after that don't connect it to 12V or it would exceed the LED max current.
If you really want to use a white or blue strip, you can desolder one LED in each string and replace it with a jumper.
You can also use a voltage booster, but I think it's a bit wasteful to boost 7.2V to 12V and then burn 6V off it in a resistor to light 3 red LEDs which only need 6V... better change the resistors so the red strip lights properly on 7.2V.