Welcome!
Answering some of your specific questions, more or less in order of appearance:
-no, spelling is not timed. It may be that your surmise that it felt too test-like explains the result, but maybe not.
-Yes, he could easily be described as exceptionally to profoundly gifted, and yes, it may contribute to his anxiety, but you also report a family history of anxiety disorders, so I would suspect multiple contributors.
-You may or may not want to explore more testing, depending on what questions you want asked, and what you will do with the information. It is absolutely possible to be both gifted and learning disabled (twice exceptional, or 2e). Data that are suggestive (not definitive) of possible LDs would include his relative weaknesses (still in the Average range) in working memory, especially the phonological loop (auditory memory), processing speed, and spelling.
Timing did affect his performance. PSI is all about speed, and he scored in the Average range on both tasks. OTOH, BD, VP, and FW are all also timed, and he did quite well on them, though it is still possible that they are low estimates, given that the other fluid reasoning task, MR, is a standard deviation higher.
-If he is happy, learning, and growing as a whole person, then I see no particular impetus to change how you are teaching him academics. If you find specific skill deficits, then work on those. For instance, addressing the anxiety with something skills-based, like CBT, might be a consideration. While many therapists may think he is too young for it, I think it would be worth a try, with a child this strong cognitively.
If, OTOH, he does appear unusually distressed, frustrated, or avoidant about certain types of tasks, then it may be worth further investigation.
