used 24G tank, should I cycle? w/ fish

[taemo]

A friend moved to BC and she decided not to bring her aquarium so she gave it to me.
The tank is a Nano Cube 24G tank, she used to have about 10lbs of dead rock, 2 hermit crabs, 1 wrasse and 3 clowns but the wrasse died days before I could pick up the tank. The clowns are 1"-2" big

As soon I moved the tank to our house I decided to clean the tank because the glass, rocks and bottom were full of algae.
I made the mistake of rinsing the glass, bio-balls and ceramic ring bag with tap water, will this be fine or should I replace them?
Also I noticed that it didn't have an activated carbon bag, should I get one and which one?
Fortunately I didn't touch the mechanical sponge nor spray it with tap water, can I clean and re-use it or should I buy a new one?

I re-used about 16 gallon of old water, the rest was fresh saltwater; I replaced the old rock with 20lbs of live rock, put the 2 crabs and 3 clowns back in the tank.
The next day we did some test and reading were:
7.8 pH
0.25 ammonia
0.25 nitrite

I know I should give the tank a proper cycle but I didn't have a choice since the tank came with 3 clowns already, how do I start the cycle? Should I put a piece of raw shrimp?
I'm on day 4 with the tank and want to do this right and not harm or worse kill the 3 clowns.
We'll be on vacation from April 4-15 and don't want to come back with 3 dead clowns and rock.
What should I do?

If I have to ask someone to take care of the 3 clowns for now, anyone in Calgary that can do that?

here's how the tank looks right now