Sure, host your website with them, but it's much more sensible (and safer) to go with a large company (godaddy etc) for your domains.
No company is without issues of course but it doesn't really make a huge difference where a domain is bought from so long as it's in your name, because the governing bodies / registries are often straightforward to contact should you run into an issue with your registrar.
It does vary by TLD though. eg, for .uk domains, Nominet are usually very helpful.
I have never really had an issue with the domains I have with Godaddy, apart from their godawful slow interface. Their old one was fine!
For .com domains, it can sometimes be harder, eg if the supplier is an ICANN member as you don't have anyone to turn to easily, whereas if, for example, the company uses Opensrs (which, by the way, 123-reg do, same as many other, smaller, companies!), there's an entire end-user management area. Of course a misbehaving ICANN member is rare, because it's very expensive and demanding to even join.
If you buy domains with a small company you're taking a risk (your domains will belong to tso/vida, not you... so if something happens to the company...)
This is categorically
untrue. The legalities alone of owning someone else's domain would be enough to put anyone sensible off doing that.
I had an issue once with a domain which was mistakenly in my name (well, e-mail address I think) - the site put up some content to which an individual took exception and I found a nasty legal letter in my inbox!