If I answer "I am at the other SIDE of the field", it marks me wrong and gives the correct answer as "I am at the other END of the field." But when I say "other end" it marks me wrong and gives the correct answer as "other side"! It seems to do this sort of thing with some frequency. Is Duolingo schizophrenic?
Etymologically speaking, I think the [l'] in [l'uomo] IS technically a short form -- or rather an elision -- of [il uomo]. In usage, there is only [l'uomo]. But [l'] is not just an elision of [il] but also [la], for example [l'uva] is also technically an elision of [la uva].
More simply put, [l'] can be either masculine or feminine. You could think of it as a "short form" of [la] OR [il], always occurring before a vowel.
So yes, [dall'altro] does agree with [lato]. First [altro] agrees with [lato]; [altro] is the masculine version of the adjective [altro/altra/altri/altre]. The form of [il/la/l'] that proceeds [altro lato] must follow the rules for agreement with [altro] not [lato]. Since it is a word that begins with a vowel, it takes [l'] and therefore [dall'altro lato].
It's a bit like the quadratic formula actually! Hope that helps and isn't too confusing.
It looks as though "l'uomo" is an elision of "lo uomo" rather than "il uomo". (http://italian.about.com/od/grammar/a/italian-definite-article-forms.htm)