cool, so your dad has a business that you will go work at to gain experience.
The staff of your dads current business wants to build a gym? What is the business that your dad currently runs?
Where will you build a gym? In the back of the current business?
why do you assume $10k? If that is the budget, then I automatically assume it will be used equipment or homemade equipment.
Think of marketing. Who is going to go to your gym? Why are they going to go to your gym? Again, location is a question you need to answer here, the neighborhood may determine the clientele and thus the style of the gym. You wouldn't put a powerlifting gym in a retirement neighborhood would you?
The big time gyms spend...well, they spend a little more that $10k on their location, building, equipment, staff, supplies, electricity, and water bills. whats your plan to pay off those bills before you start raking in the millions?
Are you a powerlifter? Cross fitter? what is your background? Whatever makes your 18yr old self special is something you should try to leverage in the creation of your gym. Power lifters own powerlifting gyms for example. American ninja warriors own gyms that cater to the training necessary to what they do.
what climate are you in?
I personally would love to lift outside. Why not a location with a big area outside that could be used for lifting?
what are you going to charge clients? Whats your plan for advertising to get clients before you open?
As for equipment, I would suggest cheaper items, shit, imagine boulders as a FREE strongman apparatus. Ropes are cheap, right? why not get some of those? TRX bands (cheap) with a knowledgable staff (maybe TRX certified) that could loosely provide guidance to clients and this could provide you with an edge over other gyms. Your list isn't awful, I just see a lack of creativity and a lot of stereotypical equipment that I see everywhere else. How about a punching bag? Those are cheap and fun and give your gym an edge.
Maybe your staff are just a bunch of gurus that provide free knowledge to members? I'm sure there would be an extra insurance waiver needed in case a member injures themselves while implementing a staff members advice.