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>I'm not sure how natural the multiple umlauts are. Any comments?

The umlauts I know have a historical order to them. In Welsh, a-umlaut comes
before i-umlaut, so that L. _grammatica_ becomes W. _gramadeg_ rather than
_*gremedeg_.

Celtic and Germanic umlaut likes final syllables, for some reason, although
there are other assimilation processes triggered by consonants; and your
changes remind me more of the latter:

e > i > y before g + vowel: _*tegos_ > _tig_ > _ty_ 'house'
e > i > y before nasal + stop, or liquid + stop: _tempora_ > _tymor_
e > a before r + cons.: _serpens_ > _sarff_
e > i > y before ss, st: _testis_ > _tyst_
e > a/o after non-syllabic u: _vesper-_ > _gosper_

>This results in 7 vowels

Not 6? Or do you have a [y] > [i] in there?