After trying many times to hear the first vowel of the verb “to play”, “i” or “e”?!...I figured out that it is a schwa, the first sound in the word “America”. So it would be məsakheq. Pealim has מְשַׂחֵק and so I think that the schwa under the first letter is the signal that the schwa is the first sound. Can anyone back me up? In addition, in the Tips section we learned that meshalem means “to pay” and I think the initial “e” there is also a schwa.
Well, the English(?) phonologists took the term "schwa" from Hebrew, but ironically, modern Hebrew lost the schwa... We're keeping it in writing (with niqqud), but the pronunciation of שווא נע collapsed into either no vowel at all or a full blown /e/ sound (not randomly; you can make quite strict generalization about which of the two in which case, and they are very consistent among speakers; but I don't think it's well-known or well-documented).
So we don't have degrees of /e/ (nor of any of the five vowels), except maybe stressed and unstressed; I don't know whether the phonologists consider it "a degree of /e/". The מ in משחק has a full blown /e/ sound. I can see why you think you heard a reduced /e/ here, but I think it's just the speaker here speaking a little bit fast.
There is one degree of freedom that I just realized: if משחק follows a vowel (like in the sentence above), it may be reduced to having no vowel: /ata msakhek/ (or /atam sakhek/, if you will). Maybe it's a little bit more colloquial. The speaker here clearly did not go this way.
Thanks! This is a good layer of knowledge to supplement what I learned in biblical Hebrew, which is that the schwa is sounded at the beginning of a sentence, when it’s the second in a pair of sh’va, when it’s under the first of two of the same consonants such as הַלְלוּיָה and after a kamats, a tsere, a holem or a shureq.