Hi, and welcome to the forum.
1. Scope input has very high impedance at low frequencies, so we can just ignore that and say it's infinite. So you measured 300mV with input Z = infinite and 250mV with Z = 50 Ohms. That tells that the output impedance of calibrator is somewhere around 10 Ohms. Basically you can solve it like you solve a resistor divider. If signal source would be 50 Ohms, level would drop exactly half.
2. Frequency matters in a sense that there is some ratio between wavelength and transmission line length. Generally if transmission line length is shorter than 1/10 of wavelength, then we don't worry as much about reflections. When calculating wavelength, keep in mind that signal speed in cables and traces is significantly less than the speed of light. Anyway, it's all relative, if you take a 1m cable, 100kHz can be unterminated, but 100Mhz should be terminated. If you take 1000m of cable, you should terminate even at 100kHz.
So should you terminate 1kHz square wave? 1kHz seems like a very low frequency, but the bandwidth required to accurately represent it is quite much higher, because it needs to pass all higher harmonics (many times the fundamental frequency of 1kHz). As you know, square wave is a sum of fundamental frequency and many odd multiples of it. Bandwith can be estimated by rise and fall time of square edges. If rise/fall time is 10ns, bandwidth is maybe 30-60 Mhz. So it's not anymore a very low frequency. If there is impedance mismatch and the cable is not super short you will start to see reflections near edges (which have high frequency content).
I don't know about a scope repair, I'm not familiar with this model.