In general, every webhost that supports PHP will run XOOPS. That much is certain - however, not all hosts are the same. I'll talk a bit about normal hosters (for normal users).
A lot depends on how their servers are set up: what versions of PHP do they use? In general, you want a webhost that upgrades their version of PHP quite often, as these updates really do benefit your site in both speed and security (or perhaps even new features).
Next to that, it also matters how the database is installed and how it's used. In most cases, the database is installed on the same servers (which isn't a bad thing - I've got the same setup on my own servers). Some hosts choose to create one huge server for their database which is only used for databases - which is also a good approach. The good thing between having a seperate server for your data and your database is that in general, you prefer having a database server with lots of RAM and some CPU, but these don't need hundreds of gigabytes of space. A server which stores data doesn't really need that much RAM, but it does benefit greatly from having a ton of disk space.
Another point that, aside from the servers, plays a role is the operating system - in general, Linux hosting is vastly superior to Windows (the main reason for this is that Windows is very resource heavy compared to Linux, and for webhosting the last thing you need is a GUI). The choice of software is also important - normal hosts often provide you with Apache, though some hosts tend to use Nginx too. The best solution, in my opinion, is to use Apache behind an nginx proxy. That way, the benefits of both are used (Apache is very powerful, but that comes at a price in terms of performance. Nginx is very fast, but isn't as good with plugins like PHP and can't work with htaccess files. It's also harder to set up).
I've read in this thread that some hosts oversell - and yes, most of them do. But this isn't a bad thing - this is just a way to make sure you use your resources to the max. If done carefully, this isn't an issue at all and you shouldn't notice it. 90% of customers never use even half of their available space and resources - it would be a waste to keep your servers at only half of it's capacity. However, there are hosts who oversell to such a degree that their service completely degrades to a very low level. This is mostly seen with budget hosters, where simple PHP scripts take up to 10-20 times longer compared to a "good" webhost.
Overselling in itself though, if done correctly, is hardly an issue and completely normal.
The biggest thing which seperates bad hosts and good hosts is backups. This is a dealbreaker, as most hosts don't even talk about this on their site - most often, it's not mentioned, and if it is, it is worded vaguely.
A good webhost takes backups for you and stores these for several days (up to month in some cases), they have the ability to restore your website for you. Most cheap hosts don't offer backups - hell, who says they even backup at all? I've seen it happen enough. And if they have a backup, most likely it's the entire machine, so that in an emergency, they can restore the machine... but they can't restore a specific file for you.
Another sign of a good host is that they have something called hot-failover, which means that when the server goes down, another one takes it over instantly. This is rarely seen in budget hosting (if at all). Even for business hosting this is rare.
The way this works is that you're either hosted on virtual machines on a hypervisor, or you're hosted on a physical server which is linked to a SAN (storage area network), which is linked to the servers. If one of the servers encounters an error, this will be noticed by other servers, which will then choose one server to take over the tasks of the primary server and will use the SAN to boot up the websites once again. This is mostly seen at companies :)
The very LAST thing which makes a webhost good, or bad, is customer service. If the service is good, you don't need them at all. In fact, some of the best hosts I have ever had had something which you would likely call bad service. Should the service go down, you'd see it on their status page, but nothing else. They focused on SOLVING the issue, not talking about the issue.
Good service is when you ask something, they know what you're talking about (in my case though, that's rather rare since I'm a Linux professional).
In terms of billing - that's a shared responsibility. Arvixe most likely uses a service to do the billing, and most likely they've marked your actions as likely to be fraudulous. Not to say that's good or acceptable, but it's understandable.
I don't think your site would go down in an instant just because your billing failed once. Most hosts give you a warning before taking an action, unless it's spamming or billing fraud. Which is something that could be arranged.
And while I might agree that Arvixe isn't the best host on the planet, it's far from as bad as you're saying here. I have had hosts that had an uptime of 80%, with an SLA of 99% and they refused to conform to their own terms (refunds).
Arvixe is afaik hosting/donating a part of the XOOPS infrastructure.
The only thing I can say is, if you really want good hosting, you're going to need to go to the high-end hosting or do it all yourself on a VPS (DigitalOcean or Vultr might be a good choice then). But keep in mind, you're responsible yourself then for backups/configuration/security - it's not an easy task. And real good hosting, which has everything that I've said in this post, doesn't come for less than €50 a month.
I myself have 3 servers running right now, 2 of them at DigitalOcean (one as a master and one as a slave as backup), the other one being my monitor and TeamSpeak server, hosted at Scaleway.