"Do you know how to speak Esperanto?"
Translation:Ĉu vi scipovas paroli Esperanton?
Can anyone explain when you use "la" in front of a noun, after a verb? Basically I'm asking why is it not "...paroli la Esperanton?"
Esperanto is in Esperanto a proper noun and therefore doesn't require la (just like you wouldn't say °la Azio, °la Usono or “the Esperanto” in English). Few other languages are nouns in Esperanto (like latino, sanskrito, papiamento, volapuko) and they follow the same rules as esperanto.
Teĥnically you could say la esperanta (lingvo) like in la ĉeĥa or la japana (so following the pattern la + adjective), but it's hardly ever used. Same goes for other noun-languages; so you can say la latina, la volapuka, la sanskrita and the popularity of these forms differs.
I've also thought that you could refer to Esperanto in Esperanto by its oldest name (used in the Unua Libro) and simply say la internacia lingvo (or just la internacia). Without a context that definitely would be ambiguous, but also understandable for most esperantists.
I've been told that in Esperanto, dead and constructed languages do not generally require the definite article, but "natural" languages do. So it would be "la germanan," but not "la Esperanton."
A similar construction was an acceptable translation elsewhere in this course.
Ĉu ni povas diri, 'Ĉu vi scipovas paroli Esperante?' (Plus, should the E be capital?)
Jes, via frazo estas tute ĝusta. :) Vi povas skribi kaj „Esperante” (per granda litero), kaj „esperante” (per malgranda litero). Pli ofte oni uzas la grandan literon, sed tio ne estas deviga.
Would it be possible to have sequential infinitives, and write this as «scias povi paroli»? Obviously that doesn't really translate into English, but does it work in Esperanto?
In this case you can't interchange freely between scipovas kaj scias povi — they don't mean the same thing. Scipovas is rather frozen in the language and thus it won't work well to split this word formation into pieces, but if anything at all could approximate its meaning, that could be scie povas (and not scias povi).
From a linguistics perspective, Esperanto syntax doesn't use verb chains as a separate construction, but since we can “feed” a verb with its object (even if the verb is an infinitive) and infinitives themselves can serve as an object of a verb, you can surely create sequences as e.g. vi devas peni esti utila al mi, kongresoj internaciaj tuj povus komenci uzadi [la lingvon], Niko ne povis ĉesi pensi pri la delikata rostaĵo, danĝere estus peni pacigi Viniciuson, nia tasko estas provi trovi kaj demonstri kio estas misfarata, la plej malfacila parto estis lerni rekoni la apartajn odorojn, mi devos kuradi alporti bieron. Those sequences aren't often and their usage seems to be limited to three verbs in a row with the finite verb (the one not ended with -i, the first one in the usual word order) being a form of esti, devi or povi.
Dankon! I did suspect that «scias povi» ≠ «scipovas», but I still can't really tell what sounds natural and what doesn't yet. I guess my main question, which you answered, was about stringing three verbs together. Obviously (as you said) it would be quite rare to need to do this, but it is cool to know nevertheless that it is possible!