Nedarim 30b - Wearing Yarmulkas

The Shulchan Aruch mentions both in o.c. 2:6 and 8:2 the idea of men covering their heads. The Magen Avrohom quoting Bach writes that one cannot walk 5 amos without covering their head, and even less than 4 amos is a midas chassidus not to go without a head covering. The Machtzis Hashekel points out that it cannot be that 4 amos is assur m'ikar hadin and less than 4 amos is midas chassidus because the Magen Avrohom writes elsewhere (91:3) that even 4 amos is only midas chassidus. Therefore, the entire idea of a head covering (for a man) must be midas chassidus, just that the bach mentioned by magen avrohom says that even less than 4 amos would be a midas chassidus which is against the pashtus of the shulchan Aruch. The GR"A in siman 8 elaborates and similarly concludes that "the rule is, there is no issur at all to go with an uncovered head, unless one is going in front of gedolim or davening then it is proper m'tzad mussar, and the same for the rest of the day for kedoshim who are standing before G-d constantly". However, the Taz 8:3 in quoting says hat nowadays there is an issur gamur to go without a head covering because of chukas ha'akum. R' Moshe (o.c. 1:1) discusses a comment made by R' Shlomo Kluger who seem to say to walk more than 4 amos one must cover their entire head (which R' moshe interprets to mean the majority of his head) and less than 4 amos must cover part of their head. R' Moshe concludes that one who wants to be machmir forRS"K should wear a large yarmulke, but m'ikar hadin one is not required to do so since even according to him the issue of covering the entire head is only midas chassidus, since the chukas ha'akum iss of the taz is certainly accomlished with even a small yarmulke.
Clearly even in the time of the gemara men were apparently covering their heads sometimes. Tosafos comments that men "are allowed to uncover their heads on a temporary basis even in front of the world, but women may not". Tosafos implies that it is more than a midas chassidus to wear a yarmulke, it is an absolute obligation, but the obigation is only to walk around regularly without a yarmulke. If one would take off a Yarlmulke temporarily for a business meeting, Tosafos would say that it is completely mutar.