If my mouth is open and my tongue is completely relaxed, do I make an [a] or an [ä]? Which one is more common for the TRAP vowel in British English?
First, I assume you're using those letters according to older dictionary practices, where "a" is [æ] and "ä" is [ɑ] in contemporary IPA-driven usage. Given the physiological conditions you describe, the vowel you would produce is closer to [ɜ], which might most closely match a UK pronunciation of "bird". Both [æ, ɑ] involve some lowering of the tongue. However, if you open your mouth extremely (all the way), you might coerce a lowered tongue, but it's not clear that you can do that and maintain the presumed "relaxed tongue". In that case, the vowel would be closer to [ɑ], not [æ] which is the "trap" vowel, as opposed to the "bath" vowel [ɑ]. However, there are two front low vowels, [æ] and [a] in IPA, so "trap" might have [æ] or [a]. There are expert reference pronunciations of the IPA letters out there – the problem is that one has to judge whether a given vowel is closer to ideal [a] or ideal [æ]. John Wells has a list of dialects and judgments on phonetic value here.