Okay for the bacterial stuff.
So when you cycle a tank (and this is me assuming you know about the Nitrogen Cycle, if you don't then skip this bit and feel free to ask about it lol) you grow a colony of beneficial bacteria. These BB are what turn your ammonia into nitrites, and nitrites into nitrates which are then slightly eaten by live plants but mostly taken out of the water by a water change. So in short, they keep your fish safe from toxic levels of ammonia and nitrites.
When you first cycle a tank, there have been products put out that is basically live bacteria in a bottle and I think that's what you're talking about. Now, these have directions on them that should be followed such as Tetra Safe Start or TSS. The directions are the turn off UV sterilizers and dump in the whole bottle to any tank right up to 30 gallons. So even if it's a 5 gallon tank, it all goes in. These products are to help kick start your colony of BB. So it's kind of like trying to populate a different plant but growing humans say from evolving other animals or would you rather dump a bottle of humans onto that plant to start your colony? Dumping the bottle is theoretically faster! So that's why you use them.
However some of these bottle contain dead BB because it's been on the shelf too long or they weren't quite alive in the first place. So there's always a risk and you really have no way of testing this to make sure they are still alive.
But long story short, if you have already grown your colony of beneficial bacteria which mostly thrive in your filter and on other surfaces (they are microscopic and you can't kill them by rubbing off the ornament/gravel/tank walls/filter/filter media, etc.) then you will always have that colony of BB, unless say if your filter cartridge broke unexpectedly and you have to replace it right away. Most of your colony is living in that cartridge because it has the most space to colonize on, so if you take out 80% of your colony, you're going to experience a Mini-Cycle. Basically your colony has to regrow again and repopulate your filter to keep converting ammonia to nitrites, nitrites to nitrates. So in that time you'll see a spike of ammonia when the BB first are removed and then in a few days they will have grown enough to accomodate your fishes waste and all (ammonia). Hence it is a Mini-Cycle because it only takes a few days as opposed to a month or about that a regular Cycle needs to grow your colony of BB.
So anyway, you can chose to use the bottle of bacteria or not. It won't help your fish to keep away parasites or anything like that. To keep parasites and other nasty's away, we do water changes to take the bad things out of the water and replace it with good things like more minerals since fish also get their minerals from water like we do
And to answer your other question, your filter should always be on because your colony of Beneficial Bacteria can be killed if you let your filter dry out, there will still be some water in there but you'd be killing off a lot of your BB by doing that. Always keep your filter running unless you need to shut it off for a water change. I always just keep mine running during a water change, but that's me lol.
There are ways you can baffle your filter too so that your baby isn't blown around everywhere. Let me know if you want the links to the two different kinds of baffles you can do!