Many prepositions have meanings, and "by" can have several, one of which is "means of transport". In (1) "sea" is considered a non-specific means of transport in contrast with "by air", "by road", "by land".
(2) "My clothes and guns were hanging in the case as well".
Here "as well" means "also", "too" or "additionally". The writer means that several items including guns and clothing were hanging in the case. "As well" is usually found at the end of a sentence, but sometimes it can appear earlier, cf. "My clothes and guns as well were hanging in the case".
(3) "It was raining heavily day and night".
No, it's not an adjective (check out the -ly ending). "Heavily" is not after the verb "be"; it is after the verb "raining". "Was raining" is the present continuous tense where "was" is an auxiliary verb and "raining" the main verb that is being modified by the adverb "heavily".