Wow.
The goal you have expressed is called "gain structure." The idea is that within in an audio system (used for making live or recorded sound), there are many places along the way to adjust the signal level, and having it high "here" and low "there" will produce the same output level as having it low "here" and "high" there, so... which one is right?
The answer is "it depends," but generally speaking, the goal is to get the input signal into a range that's good for processing and mixing, and keep that approximate signal level all the way through the processing, and then get it into a proper range for listening. Listening is usually either headphones or a power amplifier feeding a speaker, sometimes with the power amplifier inside the speaker's box so you don't have to think about it very much.
Measuring the output level from one of your boxes should be a simple matter of using an AC voltmeter that can measure the appropriate range, which is typically (but not always) 2 volts down to a small number of millivolts.
Do not worry about 600 ohms; that came from a book, based on how audio was done a long time ago -- not how it's done today. Measure the voltage with your meter, and that is that. If you really need to know input and output impedances, first assume 100 ohms for an output impedance, and then assume 1000 ohms for a balanced mike input, and 10k ohms for a line input. But to make use of those, now you have to learn about voltage dividers, and I don't think you're ready for that yet. The bottom line here is that the output level will depend, to a small degree, on what you plug it into. And my best advice is "ignore that." Measure the voltage with the output unloaded, and assume that if the level drops, you will turn a knob or slide a fader to make it go up again.
So, yes, you want a meter that measure AC voltage below 2 volts, and hopefully, ideally, to 20kHz (or beyond). While you can get by with this one particular task if it's limited to 1kHz, it will not be as good in the long run. This has mostly to do with the fact that any audio signal other than a sine wave is going to have harmonics at 2kHz, 3kHz, 4kHz, and so on, all the way up, and the things that make the sound "what it is" are going to be above 1kHz. If the meter cannot see them, any interesting signal (something beside a flute or a super-clean synth patch) will not be accurately represented. For you, that low-level AC spec, with the voltage and frequency, is obviously the most important, but understanding the rest will help you. (For example, understanding current might help you understand why you can't just keep adding more and more (unpowered) speakers to the same amplifier channel; why the amplifier maker says you can use it down to 4 ohms, or 2 ohms if it's a really good one.)
A balanced signal, which is what I think you're dealing with, has a positive version and a negative version. On an XLR connector, these appear on pins 2 and 3; on a tip-ring-sleeve connector, they appear on the tip and the ring. The signal level is the difference between those two points; that's what you want to measure. Pin 1, or the sleeve, is ground, and it is not relevant to a balanced audio signal. It's used for the shield, to help reduce noise.
I'm not sure why you introduced the word "peak." There are two distinctly different kinds of "peak" in audio. One of them is shared with electronics in general, i.e., that the peak level is sqrt(2) times the RMS level. This kind of peak is extremely rare, one might even say "never!", to use such a level in audio. It's always an RMS level, but... that level is changing all the time, because that what sound does: it gets louder and softer. So that's the other kind of "peak": when it's loud loud loud.
(There is a third kind of "peak", used by companies that make cheap amplifiers and try to convince you they are powerful enough to blow up a house... When they say "1000 watts peak power", they are saying "we are lying to you to get you to buy this thing.")
In other words, the idea of "peak" depends on the context. If we were sitting together, I would ask you where that word came from; what did you see that made you start thinking about it?
As to the specific meter, I think the most helpful thing for you would be to learn to read the specs. Look at each one, and try to learn what you need to learn to understand what they are saying. First would be the ranges; look at the voltage ranges for both DC (which might be interesting, but not for your current problem), and for AC (which is very important). So for AC, you might see "200mV", which is millivolts, the three digits after the decimal point. Let nothing go unlearned! Look for an answer for every specification, every measurement. This sort of thing will serve you in the future as well.
As to your overall project, I think it's an interesting one and will give you lots of things to learn. But at the same time, levels in audio are pretty much always subjective, especially because music and voice (and other) levels are changing all the time. It's not "one level", it's an approximate "range of levels." So getting it close is the best you can do. You want it in a reasonable range so that your equipment can work properly, so you're not overloading the inputs (too high!) and you're not getting too much noise mixed in (too low!). Then, you just see how you like it, and push it up or down as appropriate.
And it's always a matter of thinking about how to deal with it. Suppose, for example, I'm mixing a recording of a band, and sometimes I listen to it on "these" speakers, which take 0dBu for full volume, and sometimes "those" speakers, which take -20dBu for full volume. When I switch the speakers, I set the master (output) volume of the mixer higher or lower by 20dB, and everything is fine. More likely what I would do is just slide the slider until it sounds like the level I want. Generally speaking, that is NOT going to be the maximum volume the speakers are capable of producing, because my music has interesting peaks (there's that word! This is the "loud loud loud" version) of at least 10dB over the normal, average level. So now I need to make sure the average level is at least 10dB lower than the peaks, in order to keep the peaks from exceeding the speaker's maximum. And that will still sound nice and loud, because acoustic levels don't usually have to be anywhere near as extreme as most marketing departments would like you to believe.
About your audio interface, I would bet that it's pretty darn close to what they say it is. I have never measured any of mine; they just work. And they work fine. "More is more," and you turn the knob up, "less is less" and you turn the knob down. Much more interesting is how much hissy noise comes out when you turn the knob up.
If you're not doing music or other "real" sounds, and instead you're actually doing something technical, then you've got a whole lot of considerations to think and learn about! Much of the discussion changes.
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