Lernen is also used in the same way as to study in English, when you tell a child "Go to your room and study, tomorrow you have an important exam!", for example. You wouldn't use "Studiere" in this context, but "lern(e)". So I don't know why your sentence is not allowed. From my (German) perspective there is no reason against it.
Here it should be fine, but there are contexts in which it wouldn't work, like "Du lernst etwas über das Leben", which couldn't be "You study something about life", I think, or "Der Schüler lernt bei/von seinem Meister". Lernen is generically to obtain knowledge, no matter if it's by experience, personal studies, teaching etc. In this sense it is closer to "to learn".
Easiest to answer first: 1. "dass" (formerly written as "daß" before the Rechtschreibungsreform) introduces a new dependent / subordinate clause. E.g. -- "Ich weiß, dass er reich ist." (I know that he's rich.) "Ich habe erfahren, dass ich einen Fehler gemacht habe." (I found out that I made a mistake.) 2a. "das" is either the nominative (subject) or accusative (direct object) singular definite article (the) -- E.g. "Das Buch ist von J.W. von Goethe." (The book is by J.W. von Goethe."; "Siehst du das Mädchen?" (Do you see the girl?) 2b. ...or "das" can be a relative pronoun, introducing a new relative clause that has a singular neuter noun as its antecedent (Bezugswort). E.g. -- "Kennst du das Mädchen, das mit dem Lehrer spricht?" (Do you know the girl who's speaking with the teacher?"
No, that is not the reason, it is not always like that. The reason why here they say "lernen sollst" is because it is part of a subordinate clause, initiated by the subordinating conjunction "dass". In this case, the finite verb (sollst) comes always last. There are also coordinating conjunctions, they are always followed by the normal word order (finite verb second, infinite verbs last). The phrase could also be "Er sagt, du sollst lernen".