>Can you do enough frontend design to make a Wordpress site not look like a Wordpress site at all?
Yes. One of wordpress' strengths is its theme system. You can find a number of existing themes which are ready to make 'child themes' out of (I suggest looking at Responsive or even 2012.)
Bear in mind there's nothing you can do to keep someone from viewing the source and seeing it's a wordpress site.
>What's the flexibility of adding new features if I the frontend is redesigned?
You have a functions.php to build whatever hooks you need into the API for that particular theme, and plugins are basically global applications which can do anything. The API can be a bit arcane but there's a very good chance whatever problem you have has already been covered in an existing theme or plugin. I would be willing to bet that for what you describe, 100% could be handled with existing plugins, and 90% of the rest by creating a child theme and tweaking. This is honestly what Wordpress was meant for.
>Is it a better idea to hire out and let someone else pick all the plugins and put it together?
I would say it depends on the amount of work you want done, in regards to customization of the theme and custom plugins.
>Can you give me a recommendation on would could build such a thing for me, or would it be cheaper to do myself?
You could do it yourself for free, but looking up documentation and solutions is on you. How comfortable are you with horrible looking mixin PHP?
Do yourself a favor though, whatever you do: look up "hardening wordpress", apply what you learn and turn off theme editing through the UI.