I build the frontend for a Real Estate website that includes this functionality, among other things. I've also built similar 'listing' sites before, so I'll try to ask some questions that might help point you in the right direction.
It's cool that you're so open-minded to use whatever tools and tech you can find to get the job done - with this kind of attitude I'm sure you will :D
My question) Is the app and its listings 100% self-contained to your company, or will you need to fetch MLS listings for the properties listed on your site?
Why is this an important question? If you just need any app at all that can:
Then you might be able to build a WordPress-powered site that has listing plugins that work like this and be able to cobble together much of the desired functionality inside a well-documented, familiar interface.
But, if you need to interact with MLS listings (ever, not even on day 1 but you want the ability in the future sometime) that's an entirely different situation. for us it required a lot of money, as well as specific server requirements, specific security requirements, and even still as far as I know, we pay to have access delivered to 1 IP address and domain. This means even subdomains on that same IP and domain can't get the listings! That's a major consideration that could dictate what technology you're able to use, and it might rule out WordPress.
To me your Game Plan sounds right for this stage of planning. When I look at your list, it immediately jumps to my mind that you need to worry most about the design of step 1 and step 7.C, and steps 2–7 are where you will likely be able to find ready-made applications, plugins, and tutorials.
Investigate WordPress and the various business listings plugins available. Most of them will probably do 80% of what you need, so you'll want to see if any suit your needs and make modification into your final product as painless as possible.
If you do want to DIY as much listing functionality as you can, I'd recommend starting with a barebones app framework that's taken care of:
For you. These are brutally boring parts of every app but it's crucial to get them correct. Even if you wanted to write it from 'scratch' I'd still recommend starting with a known-good example of an app that includes these parts and build on top of that.
A3) Can you tell me based on the functions of this app, what skills/languages I need to learn as I go?
It sounds like you'll need HTML and CSS, and likely a bit of JavaScript on the frontend, and then you'll need a database (MySQL is reliable and ready!) and at least one language server side - this can be PHP, Swift, Python, Ruby, JavaScript (node) or anything else you're comfortable with. There are even C++ and Cobol web frameworks out there, so if you happen to have experience with any kind of language there may be a web framework ready for you to be creative with :D
A4) Which parts of this project are worth reinventing the wheel vs. finding a plugin to integrate?
Oops I kind of answered that in A2), basically anything touching user accounts, authentication, and money should be built by somebody who does that as their primary thing, and the fun parts are the parts you'll either build yourself or find ready-made block to help build your vision.
A5) Is it even possible to create a separate section of our website that is not using wordpress and it all be under the same domain? We are using Bluehost for hosting, just FYI
Yes, you could have WordPress as the main folder of your site and have unlimited subdomains, and you can also create as many folders to host static content, apps, or other WordPress sites 'within' a domain too!
…though note that if you are interacting with MLS listings and other Real Estate listings coming from other servers it may mean that you only have the money to license one address + IP, so be sure you know the URL you want before paying anybody for access to listings :D
Good luck to your on your journey, and I hope this first app-building experience opens a whole new world of possibilities for you!