So the source of electron current or battery is connected to the electrodes not the electrolyte, right?
Electrodes serve as contact to the electolyte, but they can't pass the charge without reaction taking place.
the battery is sucking electrons from anode making it positively charged and attractive to anions, the battery is also giving those electrons to the cathode making it negatively charged and attractive to cations?
Again - that's convention and you will not force me into stating "yes, that's the cathode"
But you get the process behind right.
And I just want to clarify something: the construction of an electrolytic cell is basically just two electrodes immersed in one container of electrolyte right? Because I'm not sure how else the anions can flow around either to cathode or anode. In other words, the concept of salt bridge etc. doesn't apply here right?
It doesn't matter. You can prepare electrolytic cell with two separated containers and connect them with salt bridge. It is not necessary, but it will not stop the electrolysis.
Salt bridge is necessary when you want to separate two solutions. Imagine you have solution of Fe
2+ and solution of Ce
4+. If you mix them, they will react in the solution and you will end with Fe
3+ and Ce
3+. What you want to do, is to force the charge to pass outside of the solution, to be able to use it. So you have to connect solutions electrically, but keep them separated mechanically, so that they don't mix. If you put them in a separate containers and connect them with a salt bridge they still can't react directly - even if they are eager to
- but they are already electrically connected. However, circuit is not closed yet. To close it you put connected electrodes into both solutions. Then reaction takes place - just instead of electrons being exchanged directly between cations in the solution, they are forced to flow around, through the wire, and at the same time charge (in the forms of inert ions) flows through the salt bridge.