The answer to the first question is "no", in this case it could not be omitted as it points to an object/process taking place at a time and location. The shout could be reformulated as a general warning like (actual nautical expression):
"Wahrschau Welle !"
"Vorsicht Welle !"
when it's clear where the waves come from. Else people might look in the wrong direction.
For the given example
"Da kommt eine neue Welle"
would literally translated ofc mean
"There comes a new wave"
while one could well use
"Hier kommt sie !"
meaning it'll soon have made it from there to here.
The answer to the second question is "yes" in general, but the use depends on geography. Especially in the southern parts of the German speaking world "da" has the meaning of "hier", like in:
"Komm da her" = "Come here"
"Da ist es doch" = "But here it is"
while other parts have a more distinct separation between something that's "hier", more or less at arm's reach, a little bit(tm) away "da", or even "dort", out of sight or wherever. No exact distances can be given, depends on situation, and meanings blend with time and place.
For the example of passing the plate, someone from Hannover (the standard German speaker) would say "Hier ist er" (der Teller), holding it in their hand, while in the same situation someone from Stuttgart would say "Da ist er" (Hast du ihn ? Gut festhalten).
This language use may sound unusual at first for people from the northern parts. Duden has both meanings listed: https://www.duden.de/rechtschreibung/da_Adverb_dort_hier