Note that Scottish has the contracted form “Scotch” (also “Scots”, where the use of /s/ is I think a Scottish feature).
I would guess that the consonant cluster in the middle of “English” inhibited the development of any monosyllabic contracted forms—“Englsh” is not exactly a validly formed syllable in English.
Alongside Welsh we have French and Dutch (although in Dutch the contraction happened before the word entered English).
As far as I know, there is not a general rule converting Old English short æ to Middle English e. The general rule that I have seen given in linguistic sources is that Old English short æ corresponds to Middle English a. But there are complications.
The "Welsh" word had various vowel qualities in Old English; the form "wælisc" is only a standardized representation of one form. "West Saxon" is the dialect that is most commonly used as the standardized representation of "Old English" (e.g. in dictionary references to OE words), but it isn't the only dialect that contributed to modern English.
The number and distribution of "a-like" vowels differed between different dialects of Old English; I don't know enough about this topic to summarize it, but the OED entry for the word Welsh gives the following information:
it classifies the forms into three main groups:
α. those like "OE Wilis-" (with i, y or u after the w)
β. those like "eOE Uuelesc (Kentish)" (with e after the w)
γ. those like "OE–eME Wælisc" and "ME–15 Walshe" (among others) (with æ or a after the w)
Also a minor "δ." group with the vowel "o".
in its "form history" section, there is extensive discussion of the history, including the following paragraph:
Some of the β. forms represent the Kentish reflex of early Old English æ with regular breaking to ea and i-mutation to e, and this is probably continued in south-eastern Middle English forms. However, the majority of the β. forms represent the Anglian reflex of æ with retraction to a before l plus consonant (compare the corresponding noun, Anglian Walh ), i-mutation of a to æ, and subsequent late Old English raising of this vowel to e in the east midlands and the north (compare A. Campbell Old Eng. Gram. (1959) §193(a), R. Jordan Handb. der mittelenglischen Grammatik (ed. 2, 1934) §62). The γ. forms show the expected reflex of the Anglian retracted and mutated vowel in the west midlands (Middle English a), but appear to be unusually widely distributed in Middle English. Although they become obsolete by the end of the 16th cent., Walsh survives as a form of the surname.
I think it's worth making another post specifically for the question about correspondences between OE "æ" and Middle or Modern English "e"--it seems to be a big, complicated topic, and I think making it a post of its own might attract more expert attention! I'm sure that there are other comparable examples; I'll try to find them when I have some time later.
An earlier question that seems relevant is Why "English" but not "Anglish"?, although unfortunately it doesn't seem to have gotten any answers from experts.