Is that text binding since the service was paid for, used, and taxed
in Washington State?
Most likely.
Does it matter that the web hosting service uses a mailing address in
Arkansas?
Not really. The relevant legal question is whether the web hosting service has any meaningful connections to Texas at all. If it does, that is a proper forum.
Can Bob still sue in his state (Washington) to try to compel his
refund because the web hosting service will choose the lesser cost of
a refund, or does Bob risk legal action and legal costs from the
hosting service base on the language of the terms of service?
If the web hosting company disputes the Washington State venue, Bob will probably lose, especially in a case where the web hosting company hasn't brought the lawsuit in the first place to collect a debt from him and the lawsuit is instead commenced by Bob to obtain relief from the web hosting company.
Whether there would be any downside to suing the web hosting company in Washington aside from getting the case dismissed and having the web hosting company's out of pocket costs awarded against him is hard to say. Normally, attorney's fees are awarded either based on a contract provision that says so, or because the claim was groundless and frivolous. It isn't clear if either condition would be present if Bob sued in Washington State and had the case dismissed based upon the choice of forum clause in the contract.
Also, FWIW, while the choice of forum clause allows either a state or federal forum, the federal forum would probably only have jurisdiction on diversity of citizenship grounds (there is no obvious federal claim in this case) and would only have jurisdiction if the amount in dispute exceeded $75,000 which is unlikely. So, realistically, Bob would have to sue in a state court in Texas in the county where the defendant web hosting service had its principle offices.
Bob might be able to bring a small claims lawsuit there, but given that Bob lives in Washington State and that small claims lawsuits can very rarely be prosecuted remotely, that probably isn't a good solution for Bob. He'd be better off suing in a non-small claims limited jurisdiction court in Texas with a lawyer handling it for him. And, if that isn't cost effective, he'd be better off not trying to sue for a refund.
