Yes, מתחת requires ל, but there is also the form תחת that is used directly - תחת הסוס. It's a bit more old and formal.
Without being fully positive on the topic, I could believe that מתחת implies beneath without touching and תחת is under in a way implying touching.
In parallel fashion, מעל could be above and על could be on.
I agree with you about מעל/על and it makes me think that maybe מתחת/תחת were once used in an analogous way, but I don't think they are used like this today. I think the only time I hear תחת used as a preposition it is in a metaphorical rather than physical sense - תחת השפעה etc..
In this construction, would it always be "the horse" and not "a horse?" Unpointed, it looks the same to me.
"יְהִי-דָן נָחָשׁ עֲלֵי-דֶרֶךְ, שְׁפִיפֹן עֲלֵי-אֹרַח--הַנֹּשֵׁךְ, עִקְּבֵי-סוּס, וַיִּפֹּל רֹכְבוֹ, אָחוֹר"