Regarding blade, kage and måle, I have nothing to add to Nardog’s answer, except to note that among younger speakers, the schwa-assimilation that takes place in kage is more likely to yield what would in standard IPA be transcribed as [ˈɡ̊ʰæ̝ː.æ̝], with the schwa assimilating completely to the preceding vowel. (And also to add that your informant’s description of a simple schwa, requiring belly punches and whatnot, is somewhat fanciful!)
Regarding mand and mænd, a bit of clarification to Nardog’s answer seems in order.
To begin with, if mand and mænd are transcribed with the same vowel, then the system is quite unequivocally wrong, because these are two entirely different vowels.
As Nardog mentions, there are various conventions for transcribing Danish, some of which go against each other. For present purposes, there are four main varieties to deal with: Dania; standard IPA; the ‘normalised’ IPA used, mostly phonemically, in Danish works (such as those by Basbøll and Grønnum, and also in reference works like DDO); and Wikipedia’s ‘IPA for Danish’.
The vowel in mand is [a̝] in fairly narrow standard IPA – that is, it is somewhere in between IPA [a] and [æ]. This sound is transcribed [a] in both Dania and ‘normalised’ IPA, but Wikipedia opts for [æ]. This is the basic Danish vowel that comes closest to the (British) English CAT vowel, standard IPA [a].1
The vowel in mænd is [e] or [e̞] in narrow standard IPA – that is, it is just a tiny bit lower than standard IPA [e], but not much. In Dania, this is transcribed [æ], while ‘normalised’ IPA uses [ε] and Wikipedia uses [e]. This is the basic Danish vowel that comes closest to the (American) English LET vowel.2
The vowel in æble is the same underlying phoneme as in mænd, except long: phonemically, it’s /εː/, and it’s generally transcribed as just the long counterpart to /ε/. Phonetically, it tends to be slightly more open than its short counterpart, between [e̞ː] and [ε̝ː].
1 There is a Danish vowel that is narrowly transcribed as standard IPA [a], but it’s not phonemic: it’s the allophone of the phoneme /ε/ (the vowel in mænd, see above) which is found after /r/.
2 For some older speakers, the allophone of /ɛː/ found after /r/ can actually come closer to the LET vowel than this one does, but in most younger people, that allophone has shifted and become much more open and retracted, to the point that many now pronounce it more like [æ̠ː]. And of course, unlike the LET vowel, this allophone is inherently long.