is it necessary to retain the accent when translating into English?
No, it'll accept it regardless in English. However, it needs to be added as an alternate translation (as how Duolingo works), so sometimes it's missing. – If you get it wrong due to missing accents, do report it.
Can "Szívesen" mean "dear"?
No, it means "you are welcome".
You could think of it maybe as "Med fornøjelse". I guess...
Szívesen can be used as an adverb having a similar meaning as German gern.