Are the l and the t pronounced in zvlástní? (My keyboard does not have the caron sign).
In the standard pronunciation yes, but people are sometimes lazy with the t. But it will sound very colloquial or uneducated then.
I have a question: why "zvláštni" here means strange and not something like "separated"? I know that in usual situation ,when people for example say "Dáme to zvlášť" ,they mean "We will put this separatly". Tell me if I am mistaken.
zvlášť is indeed used also for separately, independently (samostatně), but it has more meanings, for example: especially, strongly, special for this purpose
zvlátní is a bit different. First it is an adjective, but the meaning is somewhat different and again there are several meanings: strange, exceptional, extraordinary, special, weird and yes, even separate
For example zvláštní jednotky = speciální jednotky = special units (forces), zvláštní náhoda = a weird/extraordinary coincidence, zvláštní pokoj = separate (bed)room (but in a different context a strange/weird bedroom
The actual meaning depends on the context (as usual).
I think he's saying they're not silent in any situation. It's just an error in Duolingo's pronounciation.
Why is it zvláštn(i) not zvláštn(y) if we're talking about a masculine. Same goes for posledni. Why not posledny?
Those are both "soft" adjectives, and the soft adjectives' declension differs from that of hard adjectives. See, for example, Soft Declension at this handy site: https://en.wikipedia.org/wiki/Czech_declension#Soft_declension.