Welcome to the hobby!
1. For now, you want to get a hold of a test for ammonia, nitrite, and nitrate. However, your LFS can likely test these. In general, you should be seeing 0.00 on an ammonia test (these can be hard to read but your lfs should be able to help), nitrite 0.00, and some nitrate at the end of cycling a tank. The easiest way to cycle a tank is to get a bottle of cycling bacteria and some ammonium chloride for fish tanks. It will take about a week or 2 before you can get fish if you start with dry rock. You will see an ammonia spike, then nitrite, and then nitrate (check nitrate after nitrite is gone). You then do a water change to bring nitrate back down. Adding live rock will make the first few months to a year easier, but more difficult in the long run due to pests.
2. No, not until you get a lot of coralline algae or stony corals.
3. Yes. You will find that an rodi unit will be very handy later on as my 75 gallon evaporates about a gallon a day. There are some cheap ones if you'd like me to share some.
4. Some companies make hang on back refugiums (fiji cube, aquamaxx, etc.). Icecap make a hang on back algae scrubber too. If you want a prebuilt AIO filter box, check out the fiji cube inserts. I will say though that sumps are far easier in the long run due to convenience. Refugiums are very helpful
IME live rock has far fewer long term problems compared to dry rock. Live rock will lead to a more stable, more quickly maturing, healthier, and more interesting tank. With dry rock you are more likely to get Dinos, have nutrient problems/imbalances (sponging/leaching of phosphates, denitrification), and have nuisance algae issue. The only advantages to dry rock are the ability to create more elaborate scapes and the lack of biodiversity (though IMO that is also a negative, the threat of ‘pests’ is wildly overstated, biodiverse tanks are more stable and more interesting). It’s really perplexing to me when people recommend dry rock as some sort of easier or less complocated alternative to live rock, my experience is completely the opposite.
OP - In addition to the beginning Ammonia, nitrite, and nitrate (for the first two, API is fine, for Nitrate I would recommend Salifert, RedSea, or the Hanna HR checker), if you’re going to keep corals, you’ll also need Alkalinity, Calcium, Magnesium, Phosphate, and PH (if you want recommendations, feel free to ask, but in general get the most specific, highest resolution tests you can, and avoid API). A lot of people make the mistake of not prioritizing test kits before buying corals, and it generally never ends well.
I would really urge you to get your own RODI unit, depending on an LFS is both inconvenient and risky. You never know if and when the LFS is changing their filters/membranes, and in an emergency being able to make water can be lifesaver (what if you need to do a massive water change outside of business hours?). I just wouldn’t trust such an important building block of a healthy reef to an outside source.
As far as the refugium, a HOB would work just fine (I have one on my 28g and it’s excellent). Im not sure what you mean by AIO. Are you wanting to add some filtration chambers to the tank? If so, I would instead recommend drilling the tank or adding a HOB overflow, and setting up a proper sump/refugium. If that’s not possible, Fijicube sell inserts for tanks to turn them into AIOs, Im pretty sure they just came out with a 75g version.
I would recommend watching BRS’s 52 weeks of reefing series, they break down all the basics in an easy to understand way. My only other advice is to just go slow and research everything before committing to anything (be it equipment or livestock)