Languages aren't a good example because we say just "Nie znam angielskiego", meaning literally "I don't know English" and translated as "I don't speak English". In Polish saying even "Nie mówię po angielsku", normal in other languages like English or German, sounds strange. In Polish for languages there is usually used "znam angielski, niemiecki, francuski, rosyjski, włoski, polski" or "nie znam angielskiego, niemieckiego, francuskiego, rosyjskiego, włoskiego, polskiego".
We usually also don't say "jeździć na koniu", but "jeździć konno".
"Nie mogę" is the most general. "Nie mogę pojechać samochodem" - "I cannot go by car". It's not important why. Maybe it's broken, maybe I am drunk, maybe I have no driving license, maybe there are some road works on the only route.
"Nie jestem w stanie" - suggests that I cannot do it because of my current condition or abilities, or even some external factors, but rather not permissions given by other people (although it's not very strict).
"Nie POTRAFIĘ dzisiaj dojeżdzać pociągem na zajęcia." is a very strange sentence. It would mean that somebody don't know how to use trains and have to get to classes for example from home many times during one day. "Potrafię" is for knowing how to do something. And "dojeżdżać" is something like "be doing something" in English.
"Nie mogę dzisiaj dojechać pociągiem na zajęcia" or even "Nie jestem w stanie dzisiaj dojechać pociągiem na zajęcia" - then it might be for example because railway workers are striking and trains don't go today. Or there is any other factor which stopped trains and caused that I must find another mean of transport or miss the classes.
"Nie mogę pojechać samochodem" - "I cannot go by car". It's not important why. Maybe it's broken, maybe I am drunk, maybe I have no driving license, maybe there are some road works on the only route.
And I forgot about one more case possible - there is no person who would give me a lift. In Polish there is no difference between "go", "ride" and "drive" (if talking generally about using a car for transportation, not just sitting behind a wheel and steering it), all three are translated as "jechać". Maybe except "go" as "walk", then it's "iść" instead.