I’m sure @DaveT or @howard can give a better answer if that didn’t help.
Yep. The gain on the input is not like a volume knob that reduces a signal to a level that works. It’s a variable amount of amplification to apply to the input signal. If the input is higher than the preamp is designed for, the gain isn’t going to fix that.
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As I understand it, gain can only increase the input level, but not decrease it. Would that be a correct statement?
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Yes.
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As @akos pointed out, it’s not at all necessary if you have a ground lift button because that does the same thing.
Without a button I’d make my own ground lift cable before I started buying noise isolators and the like.
Yes, in that other thread they were using a cable that had pin 1 also tied to the connector shell. Even though they punched the ground lift button, the system was still forming a ground loop through the shell making contact with a metal chassis. In that case the only way to solve it is by clipping the jumper in the cable between pin 1 and the shell.
In a permanent pro install, all the balanced cables only have the shield connected at one end. The noise on the shield only needs to be drained to ground at one point. Any additional connection leaves a ground loop possibility.
I was writing show control software for a stage production once where it was convenient to sit next to the mixing console because I could plug my laptop into their power strip and have a good view of the whole stage. My laptop was connected to the show control computer with an extended RS-232 serial data cable to upload code revisions. I was so focused on what I was doing, I didn’t notice the audio team was running around trying to get a ground hum out of the mains. I created a ground loop between tbe audio system and the show control system through the ground pin on the RS232 serial connection. They were relatively forgiving. The point being that they were able to find the ground problem pretty quickly only because they had the system already wired with all the grounds only tied at one end. Otherwise, it would be a sleuthing nightmare.
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This must have been quite a while ago 
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Technically correct, but many knobs labeled level, gain or master are actually electronically attenuators. The amp stage often runs wide open all the time at full gain and we turn down the signal we are feeding it to get less overall gain.
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Yes, these are early career lessons, not to hose the audio system with my laptop power supply. I was synchronize starting a stack of Pioneer laser disc players.
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It’s less hot, but maybe not less enough.
You get one turn of the knob to make a useful adjustment. If it’s a mic input gain knob, it will give you the ability to adjust to a range of mics. It may change the overall gain of the input stage from 30 dB to 50 dB at the ends of the knob turn. 30dB of gain is still too much for a line level input. Mic/line inputs will have a button for the line level option (or software switch) that kicks in a pad to lower the line level down to the top of mic level range so it won’t overdrive the input.
All of these are arbitrary choices by the circuit designer and how much budget they have.
High gain mic preamps are more expensive. So a less expensive mic pre may not have enough gain for a particular ribbon mic, for example. That’s why you can buy special ribbon mic preamps or people use another gain stage like a Cloud Lifter.
Using devices with meters helps a lot. You need to know how hot you are running. Guessing by turning it down when it distorts is a lot more difficult and inconsistent. This is also why people end up with noise, not running as hot as they could be.
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Actually it’s always the case, more or less. So a gain knob goes from 0% to 100% of the maximum voltage gain of the gain stages. That’s how you can have “less than unity gain”, somewhat. Just another way to say the same thing.
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I think this is the concept that’s confounding @Malyngo. He wonders why you can’t set a mic input to 1 and have it be a line input.
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yeah, that’s why I wanted to put some other words on the same things that you already said 
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Yeah on reread it looks like my answer was kind of ambiguous for what he was actually asking
You actually can and do attenuate input signals down to the lower levels (in the same way that pad switches or DI boxes do), but there’s no guarantee a gain knob would let you do so enough or have enough of a useful range after doing so.
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