For clarification: 5 years ago Mechatrommer was a starving student putting his valuable software (eg: Goltech for the Hantek DDS3x25) out there for the rest of us for free. I don't have those skills, I just leached off his generosity. But I did have a shared webhosting account at Dreamhost and it didn't cost me a dime to make space for soasystem.com, saving Mecha a few bucks. Win-Win.
I have been using Dreamhost since 2005. I think the price is the same now as it was then -- about US$190 for 2 years if paid in advance. Includes 1 domain name. Some domain names have gone up since then, I think additional names were $9.95 originally, now there's a sliding scale based on the top level. You can still get something like a .eu for $10, but .audio will set you back a cool US$130! The included domain name has to be one of the cheaper ones now.
I chose DH originally for myself because it was the best bang for the buck. You get PHP, CGI/FastCGI, unlimited number of MySQLdatabases, unlimited bandwidth, unlimited storage, unlimited visits, unlimited e-mail addresses, unlimited (and free) subdomains, SSH access, automated Let's Encrypt free SSL certificates, cron, automated back-ups, etc. I run several different informational websites, but have no experience setting up an e-commerce site. I have not looked at what is out there lately because overall I have been satisfied with Dreamhost but comparing DH regular price to Hostinger's regular price, DH is still the better bang for the buck IMHO. And both are way ahead of GoDaddy. (Somebody's got to pay for all their expensive advertising! <lol>)
I can say that there should be no problem with anything you mentioned. For Joomla, see:
https://help.dreamhost.com/hc/en-us/articles/360047925992-Joomla-Installing-and-updatingFor Zen-Cart, see:
https://help.dreamhost.com/hc/en-us/articles/360045866891-Zen-Cart-InstallingThere are a wide array of other options, too. I personally use Drupal for most sites.
If you are going to be taking payments through your website, an SSL certificate is essential. As noted above, DH makes it painless. One-click installs of the free Let's Encrypt certificate, you can force https only access with a second click. You can edit the Apache .htaccess file to force TLS1.2. That said, you might want to fork out the $15/year for a paid certificate since you are setting up for e-commerce. See:
SSL certificate:
https://help.dreamhost.com/hc/en-us/articles/215089118-Adding-an-SSL-certificate-overviewUnlike many others, Dreamhost does not use C-panel. They have their own custom interface but it performs pretty well all the same functions. e-mail support is free, telephone support costs. They used to offer a wider array of automated "one-click" installs that provided automated updating. At one time, that included Joomla but now they seem focused on WordPress almost exclusively. I'm not a big fan, WP is probably the world's most hacked website software, so I don't use it.
You are welcome to stay were you right now are for as long as you like, of course!