Taking your second point first. Most test sources have a fixed output impedance of 50 Ω, and your AFG3000C is no exception. From the datasheet you linked to (Amplitude table, page 4), it says ...
Output impedance: 50 Ω
There are several reasons for this
1) It's not possible to achieve zero output impedance over any useful bandwidth, and if you did, you couldn't maintain it at the end of a cable
2) With a cable and generator of the same impedance, the output impedance at the end of the cable is the same as that of the generator
3) Many things at high frequency require a finite impedance to be stable, when testing RF transistors for instance
4) As a result, 50 Ω has become the de facto standard impedance for cables, connectors and test gear, which makes it very easy to connect things and have them just work.
5) Providing a good 50 Ω is enough of a struggle that no general purpose test gear provides a freely controllable output impedance
6) If you want a specific output impedance for your particular test setup, it's easy enough to use an external resistive pad to provide it.
As a result, the output voltage is usually calibrated so that it displays the voltage or power into a 50 Ω load.
Now to your first point. Your AWG goes one better. Your datasheet, same table, says ...
Load impedance setting: Selectable: 50 Ω, 1 Ω to 10.0 kΩ, High Z (Adjusts displayed amplitude according to selected load impedance)
If you tell it what the load is, it will work out for you the correct voltage scaling to display. The corollary to that is that if you tell it the wrong load, or if your load is varying in a way you don't anticipate, then it will display incorrect numbers.
A point worth reiterating about the fixed 50 Ω output impedance and cable impedance. Even if you are driving a non-50 Ω load, the AWG+cable combo still looks like a 50 Ω source at the end of the cable, and (neglecting cable attenuation) the voltage display on the generator is still valid at the end of the cable, regardless of your strange load.
Your third point. You'd like there to be an 'absolute maximums' section, which specified what the highest external signal that could be applied was. I used to design signal generators, and we sweated bullets so that it would survive if somebody accidentally keyed-up a 50 W transmitter into it. However, all your datasheet says is ...
External voltage protection: To protect signal outputs against external voltages use fuse adapter 013-0345-xx
So I guess it's fairly fragile. Don't reverse drive it with any signal, and buy the fuse adapter if there's any likelyhood of you doing so. You could ask the manufacturer what the specifications of 013-0345-xx are, and how it improves the survivability of the generator.
Always read the data sheet carefully for details of the reverse driving specification. Some instruments can be particularly fragile, some can be surprisingly tough (like my signal generators). You can't tell by looking at the instrument, or at the price tag. You won't know by experiment, until it's too late (my favourite phrase from a high energy forum I use is 'turn it up until it catches fire, then back off a bit'). Your AWG says nothing about survivability, except to recommend an extra protection circuit if you're likely to reverse drive it. So don't.