The food sounds good, she may start eating again once everything else settles down. A 10-gallon is good for how young it is, but once it passes 5 inches or so, it's best to put them into a 20 gallon long/wide tank to allow them to walk/swim around comfortably.
Room temperature only works if your house is in an extremely consistent low 60s range. The fan is a good thing and may be enough, but you might have to purchase something more heavy duty in warm months as the temperature absolutely should not exceed 70 degrees F. You also want to avoid major temperature fluctuations (more than 4 degrees in less than 12 hours) if at all possible. I would say that you probably don't have to worry too much about the temperature being too low as long as it doesn't get below 55 for too long. Your ideal temperature range is 62-65. I typically aim for 62 myself.
I often leave my fan running all the time, but if you're concerned, the temperature controller should also work with the fan--you would just leave the fan switch in the on position, plug it into the controller, and that will turn it on whenever it's needed. Sorry I haven't gotten the link yet, got distracted, it'll be here soon.
The tank absolutely needs to be cycled first. Axolotls produce a ton of ammonia-producing waste and as amphibians who absorb everything through their skin, they're relatively delicate. The people in that shop are very VERY wrong on that count. I would say that you should definitely tub her until the tank is cycled. I personally had a lot of success using liquid ammonia (you can get it from Walmart or Amazon or lots of other places). There's lots of great guides on how to cycle a tank online. A liquid water testing kit is an absolute must for this project since you need exact information.