Translation:The former train station is a restaurant now.
Yes the adverbs are definitely synonyms, but the adjectives can be synonyms only sometimes. Occasionally, "former" would not always be able to replace "previous". For example on a page, you can talk about the previous line, which would be the line before the one you are on, but "former" would mean that you have changed the current line and it used to be different.
They are not identical in meaning. Previous is something that was existing or occurring right before something's current state, while former could be what something had existed as at any point in time in its past.
An example could be if this hypothetical restaurant was previously a general store, though before that had formerly been a trainstation.
Because the definite article (der) already takes the duty of 'strong declension', so the adjective gets the 'weak declension' ending: http://www.nthuleen.com/teach/grammar/adjektivendungenexpl.html
Interesting. If this is common usage for America, you should report it. A train depot here in the UK would probably be taken to mean a shunting yard, not somewhere for the public. In the UK, it is common to speak of a railway station, not a train station; and equally common to call it simply a station because railway is generally taken for granted.