booking Input needed - Riding Stable site w/ online booking for rides/horses

I need a little input on what I think will need to be either a a Booking site with restricted access for who can book and what they can book or perhaps even a Member site with varying permissions for bookings. 

(I've mocked up a rough sketch and included as an attachment)

Context:

I've only ever built sites for creative businesses (more portfolio based) but have been asked to build a site for a Riding Stable that needs to streamline their booking process (the owner is doing it all manually right now via text and Venmo).

I almost picture it like a Salon booking system...where clients book with certain stylists, and each stylist has their own availability.  Just in this case, the "stylist" would be the horse?  So if anyone here has built a salon website with this functionality, I could probably piggy back off of that concept if you'd be willing to share. 

Necessary Parameters:

• Only active/current students can book a lesson
  • (so the booking area needs to be restricted somehow)
     
• Students are only able to ride the horses that fit their skill level
  • (so booking must be restricted to certain horses)
     
• And those horses are only available certain days/time
   
• some students pay monthly, and some pay per lesson...
  • how would you differentiate this in the back end? 
  • perhaps making "pay per booking" the default, but then give students who pay monthly a promo code?

Additionally:

she needs a way for people to pay for horse camps (both deposits to hold a spot, or in full after a certain date) and merch, but I think that I can easily handle by just setting those up as products, no?

The rest of the site will be your usual:

• photo gallery
• "team" page (which will really be the horses and their descriptions)
• blog/news/updates page that I'll likely feed into the home page with a summary block
• contact form
• about. 

I'm good with all that, but just pointing in out in case there's anything planned that wouldn't mesh with the above. 

Style/Template Help:

she was not specific, but she's currently using a one page Godaddy site now, so anything will be an improvement, but overall I'm picturing just a simple top header image or slider, a few sections below the fold with "meet the trainer", "tour the barn", and "meet the horses", top navigation to the pages mentioned above, and footer with contact info.  

Thank you for any suggestions and input!

Lastly, as this is the first time I'm undertaking a job like this, how would you quote it?   
I generally like to do a flat rate that includes "X" plus hourly for anything over and above that, but I still like to base my flat rate on roughly how long I think it will take me.  Anyone built anything like this who can give me a rough idea of the number of hours I should budget my time for this project ?