HEY THERE! WELCOME.... i agree with purple turtle completely good job asking for advice! Your in good hands mate!

1. Does the rock at the bottom need to be replaced, and what is it?
NO need to change out the rock bottom. This is used as a BIO FILTER. By the looks of it it is the EXO TERRA BIO DRAINING SUBSTRATE! I USE THIS AS WELL !! =)
Here is more INFO about the EXO TERRA BIO DRAIN SUBSTRATE!

2. Fred told me the temperature needs to be between 60-85? He said room temperature is fine as long as I'm misting. Can I just get a spray bottle for the misting? Fred said it would be fine but I want to make sure.
Yes Fred is right! I would say closer to 65 but shouldn't be an issue going to 60 for a shorter period of time! =) yes you can get a spray bottle... MIST 2 times a day and you should be good. =)
3. Should I get a temperature and humidity gauge? The man at the conservatory told me it isn't necessary, he said as long as your home is room temperature, the combination of misting and the setup of the tank (sphagnum moss etc) should keep the humidity just fine.
I would get it just so you know how much to mist your little guy. You may mist 2 times a day but that doesnt answer your question "HOW MUCH IS ENOUGH". I would get it yes. Keep your humidity between 65-85 percent =) I do that my for guys. I maintain usually around 70-75 percent.

4. Do I NEED to coat the crickets in vitamins? I'm feeding them carrots, which he said should be fine too.

YES!!!! I would be very carefull on vitamins...they will only need that about twice a month...TOO much of it and your froggy friend will pass away. You do HOWEVER need CALCIUM!! NO QUESTIONS THERE!! Dust your crickets Ever other feeding with Calcium. Make sure the Calcium has D3 in the product. Heres information on D3 Calcium!

I hope this helps... I am a breeder of these frogs and there are many people on here that know i do my best to care for them...INCLUDING the overly expensive vet visits haha. Well Best of luck...feel free to private me if you ever have any questions mate =)

-COREY