Is it possible to have multiple of each?

Yes.

 

One of the most common methods for doing the eggs fast is to camp Barb fish (since high hp per hour rate = high prawn ball/cracker rate) until you acquire all 3 of their eggs plus 11 un-opened mystery eggs and/or gold crackers then open the lot and tada all 14.

Yeah, found a picture of someone with loot on Reddit from 500 prawn balls, as you can see, multiples:

 

Yes.

 

One of the most common methods for doing the eggs fast is to camp Barb fish (since high hp per hour rate = high prawn ball/cracker rate) until you acquire all 3 of their eggs plus 11 un-opened mystery eggs and/or gold crackers then open the lot and tada all 14.

 

If you do not acquire 11 unopened mystery egg/gold cracker after getting the 3 leaping fishes, do you move onto a different fish or continue to barb fish?

Keep your unopened mystery egg/crackers (don't open them yet) and proceed down the list of most efficient fish in terms of harvesting rate for prawn balls. You get a prawn ball for every 25K LP worth of fish caught. Barbarian fish are assigned some value even though they aren't actually consumable. After that, the list goes (courtesy of the Wiki page):

 

Monkfish

Swordfish/tuna

Lobster

Salmon/trout

Shark

Lava eel

Rocktail

Cavefish

Rainbow fish

 

Keep going down the list, progressing to the next spot when you obtain all applicable golden eggs in one spot, until you have enough mystery eggs to cover it.

 

Note: I have assumed that your primary goal is to finish the Aquarium as fast as possible. If you would rather take your time and enjoy the fishing xp, continuing at Barbarian Fishing all the way through is probably a better alternative.