"The sales person said"
There's the first problem. Be careful, be skeptical. They're not interested in making the world a better place, mitigating climate change, or giving you cheap energy. (Well maybe some of them genuinely are). They're interested in getting your money.
"Question 1: Is this correct? That during the day I effectively have "free power" (not including the cost of the system)."
Well kind of yes, but "free excluding the system cost" is a big caveat, and you only have so much power up to a point, and it's not completely reliable and predictable when you want it.
"Would it be feasibly to have a UPS that charges up during the day and powers the server overnight."
In principle yes, but what's the watt-hour capacity of your UPS? Mostly they are only designed for a short run, enough to safely shut down and power down the downstream systems.
"Question 2: Am I going to burn my house down? The wife would probably be annoyed by this."
Not if you have reasonably good quality gear, quality wiring etc.
"Question 3: Given I am going from DC solar -> 240 AC -> DC battery -> 240V AC, is this efficient enough to be worth it? I am effectively looking at if I put power onto the grid and then access it at night, that power is ~30% efficient, because I only get ~30% of the price to sell the power vs buying it. To put this another way I gain 18.4c/kwh by storing the power and using it versus sending it to the grid."
Direct DC use at the appliances is more efficient. Many big datacentre systems and companies like Google are seriously getting into this concept with 48V distribution to all the servers.
But is the time, labor, engineering that is going to be needed worthwhile for a small-scale system?
"Question 4: Is this a terrible idea? I can't find anyone else asking about it, which makes me weary."
It seems like it's not bad as an experiment in small-scale, low-cost, "try before you buy" home battery storage.