Is this incongruence problematic for branding?
That's a context-dependent measure I'm afraid. Some brands it might work perfect, others not so much. For example:
Southern Expert Xerox Yogis —> sexy.com while other names like Prescott
Williams Murphy McMahon Williams & Schroedinger Attorneys at Law —> wmmwslaw.com would probably be a lot more user-friendly.
Which should the website logo reflect, PVGA or the full business name?
Again, it's all about context and application. On the website, there's not much SEO impact of adding a new logo, though you probably want to stay away from changing the homepage title too often. That should really only be done a lot initially when you're still playing around, or down the road as a very concerted re-branding effort.
There's a lot of stuff like meta tags and publisher information that you can add to help keep SEO organizational attribution static, and are more actively consumed by search engines as well. For example, the JSON-LD convention:
"name": "Your Business Name",
"description": "A description of your website",
"name": "Your Brand",
There's really not too many SEO concerns when it comes to actual domain name, even for obscure tlds like
.kitchen or etc. It's really more a matter of UX and brandability.
xjys.contractor looks sketchy compared to
xjys.com but both look off-putting compared to (IMO)
xjyardservice.com. I think it's all about the site audience, their preferences, and ultimately the impression the name gives them. Indirectly, I suppose that could have an effect on UX-related site interactions like bounce rate, organic CTR, etc. but that'd be guesswork and hard to measure in any practical manner.