Is it okay to make a loop of LED strips?
If you connect your positive supply cable to one side of the LED Strip and the GND cable on the far end of the LED Strip the LEDs will be the same brightness.
A antenna for a 1 KHz signal must at least be 75km long so i would not expect any problems in terms of radiated emissions. Since you are using a square wave you must consider the slope of the signal though, which you can control by controlling how fast your switch turns on (eg. with a gate resistor when using a MOSFET).