Welcome to the deep end!
Not in any way to discourage you, but rather to inform, so you know what to be on the lookout for and can plan ahead...
(TL;DR: your tank is not big enough for a clown and a dottyback together, turbo snails are too big for your tank, and take it slow on adding corals)
Speaking only for myself, I would be at least a little nervous about a clown and a dottyback in a 15-gallon tank long term. Both fish can be territorial, and dottybacks in particular have something of a reputation for being spicy little jerks once they get established. In a bigger tank, that can sometimes be managed because each fish has room to claim territory and avoid the other, but there is nowhere to really escape in a 15 gallon. If one starts harassing the other, the loser can wind up in a pretty constant state of stress with nowhere to get out of sight.
Like, I seriously don't mean to be the sort to shout "doom", nor do I want to be the bearer of bad news, but I've looked and looked at multiple primary sources, as well as asked 3 4 different LLM's - ChatGPT, Google Gemini and Perplexity.AI and Claude - all while being VERY careful not to bias the results with my search terms/questions, just to make sure I wasn't off-base in my immediate reaction, and I cannot find anyone or anything that thinks that pairing of fish in a tank this size is a good idea. I'm frankly kind of shocked that the people at Pure Reef let you walk out the door with them if they knew anything about your setup.
That same concern can extend to the shrimp, too. Not every dottyback is a cold-blooded murderer, but they absolutely can be hard on smaller inverts, and there is not much refuge in a tank that size. Even if the two fish manage to establish some sort of stalemate, if that shrimp finds a safe hole, you may never see it again except when lights go out.
Honestly, it is just harder to stock a tank that small effectively than a lot of people realize. You might get lucky and get away with this pairing, but I would suggest paying close attention, especially as the fish mature, and being prepared to isolate or rehome one of them quickly if things ever go south - the chances of which I'd personally rate as being very high sometime in the next year or so. If they make it to ~2 years without fighting, you're probably fine. I definitely won't say it isn't possible that they could get along - I've worked with too many animals for that - but I'd give it better than even odds they won't.
Also, FYSA, turbo snails get surprisingly large - easily the size of a golf-ball, on the small end, up to tennis-ball sized! - and in a tank the size of yours they can become absolute bulldozers. I think they're a bit much for a 15 gallon personally - too big, IMO, for my 50-gallon - so I would be prepared either to rehome them later or to make sure every frag is glued or epoxied down very securely. I'm surprised Pure Reef sent you home with these as well, to be honest, at least not if they knew all you had was a 15-gallon AIO.
For a tank that small, I'd usually lean more toward a lighter, more appropriate cleanup crew than turbo snails. Something like 1 nassarius vibex - maybe 2, max - or ~4 dwarf nassarius, plus some dwarf ceriths. Then if you can get them, collonista or stomatella are both great little utility snails for nanos (and can usually be readily sourced from other club members here). Finally, maybe 1 trochus, or 2 max if the tank already has enough film algae to feed them both: if not, I'd stick with one for now, and maybe keep a small amount of invert or algae-grazer food on hand in case it needs supplementing.
Only thing to keep in mind is nassarius snails are scavengers, not algae eaters, so they help with leftover food and turning over the sand more than they do with film algae. Everything else listed will accomplish most of what people usually want turbos for, without turning the tank into a turbo-snail bowling alley. Also, nassarius are climbers, so a lid or screen top is a must anyway, just like it is with clowns.
I know a lot of folks just do not have room for anything bigger than a nano, and that is totally fine, but it does mean you started at one of the harder ends of the hobby. Reef systems have a lot more moving parts, and the smaller the water volume, the faster things swing. So regular water changes and stability matter a lot here, and stability in a smaller volume can be a much more challenging task than it is in larger tanks.
Also depending on your coral load, you may or may not wind up needing to dose alkalinity and calcium any time soon, but you still want to keep a close eye on pH, alkalinity, nitrate, and phosphate, and go ahead and have either Kalkwasser, All-For-Reef, or a two-part solution on hand for when you need it.
Still, with these and everything else, focus less on chasing target numbers, and more on keeping things as stable as possible: if you need to hit a different number, get there slowly ("slowly, slowly, catchee monkee").
In the same vein, your tank is new. You're going to have a LOT of "ugly phases" over the next 12 - 24 months, especially if you didn't do an extended lights-out phase before introducing fish and corals. If you haven't added copepods, order them NOW, along with a bottle of phytoplankton to feed them with (you'll only need maybe 7ml per day, so a single 16oz bottle will last you a good while), and be prepared to add another bottle of pods in about 3 - 6 months.
Between pods, not overfeeding, and not blasting lights too bright, too soon you can definitely shorten the ugly phases, but they ARE going to happen. Stick with things that work slowly, like adding some decorative macroalgae, or accepted-safe biological/bacterial levers like collonista and stomatella snails, and things like Microbacter Clean.
Slow down a bit on the coral additions though, and keep in mind that coral warfare - both with stinging sweeper tentacles like your hammer coral will extend, and chemical warfare via allelopathy - is a very real thing, and the effects are magnified in your small tank: I've seen my favia corals extend sweepers easily 3" long, and that 6" diameter is close to a quarter of your volume. There's also far less water in the system to thin those chemicals out before they start reaching toxic concentrations. You need to give them time to adjust, and make sure they're settled and not competing with other corals for nutrients and space. Add too many, too fast or too often, and when something goes wrong ('when' being the operative word - it's a learning process, and if it were easy everybody would do it), it will be that much harder to know just what happened. Not to mention, each and every coral brings its own needs and impacts on the system, and if you just keep adding them all the time, you'll be continuously chasing those.
Finally, in case you haven't heard it before, let me leave you with the single most valuable and important piece of advice I've ever been given in this hobby: "Nothing good happens fast in a reef tank."
(Again, welcome, to the hobby and to the forums! I wish you nothing but success, and we're here to help if we can!)
!! So thank you! I think I just got permission to upgrade!! 
