It may not take long this time and yes, no sudsing...lol...
What is your stocking plan on the 5gal?
No you don't have to use carbon, it is more of a personal choice-I don't use it-but I always have some on hand just in case I need it for an emergency....lol.....water change will do the same thing as carbon for most common issues like odors, tannin, medications, yellow water type things
Most of the carbon on the market are low quality and are used up in 2-4 weeks and need replaced-some debate still going on with used up carbon left in the filter and it leaching all the bad stuff back into the tank-I don't agree-when I used store bought filter media that was combo sponge/carbon- I have left them in my tank for over a year without any problems.....
If your filter uses separate sponge and carbon I would look at replacing the carbon section with bio-balls/media that stay in the filter for ever and provide a good surface area for the beneficial bacteria-they only need a rinse/swish every so often so that they have good water flow just like the filter sponge-lack of oxygenated water and the beneficial bacteria can die.