Sharyn,
Now i am getting more of picture of your requirements, regarding your query
I envision my Moodle somehow manifests itself as a professional looking website. On that front page are blog
articles, videos, documents, links to moodle courses, links to
discussion topics, games or links to games, social media feeds showing
on the home page, hyperlinks to external sites, articles, quizzes, etc.
None of these things (perhaps apart from the courses) need to be
assessed - they are just informational as per a typical website.
Yes all possible, you can design your front page as you like, there are templates/themes available, both free and paid, but the community inputs and feedback on free templates available is far more superior to those available for purchase, themes like Essential, elegance, Adaptable, Eguru, Academi, evolve-d etc. etc. can give you a combination of professional looking website. you can learn more about themes as per their download stats from plug-in section (https://moodle.org/plugins/stats.php?plugin_category=3) his link will provide you with themes as per their popularity. You can even have a look at moodle's own site, it's not even making use of any of the listed themes but those that comes bundled with moodle, and yet there are lots of things on main page.
I have had a look at your hosting provider and they indeed are shared hosts (with ability to provide dedicated hosting as well), by shared host I meant, one hardware hosts multiple user accounts/sites, so essentially you are sharing the resources with all those sharing the server, so customized installations (like Big Blue Button) wont be possible and even if some hosting providers can do it, the live sessions recorded are converted into accessible videos and that process requires a heavy punch in terms of processing power, that is why I recommend going with a VPS at least, or dedicated.
Regarding payments, I can link further plug-ins like classic pay (https://moodle.org/plugins/enrol_classicpay) that can take multiple forms of payments.
Members
can access some content for free and register/pay to access other
content with different membership structures. People can sign up for
free to a weekly newsletter delivered via email. They can subscribe to
my info via RSS feeds. They can be notified when new content is published on the site or subscribe to articles for the same reason.
Regarding limiting resources to users, YES, you can surely limit some resources to some users (like guests can access some parts of the course while paid students can access some or all)
and finally, you can post a job in Moodle jobs (https://moodle.org/mod/data/view.php?d=54) listing all your requirements and I am sure someone will be willing to assist you in designing your site like the way you want it, though I'll recommending giving a try oneself before you look out for help, as you mentioned you are into IT yourself and community forum is here to help you fully.
and one final advice, as you have came back to moodle after sometime, teaching with moodle is currently going on, its a free course by moodle, and I took it last session and have learnt a lot. you may want to take part in it, you'll see big blue button working there. though enrollment started on January 2nd, but still you will be able to get most out of the course.
https://moodle.org/calendar/view.php?view=day&time=1483297200#event_118100230