First of all, how old are you? Men's voices continue to develop well into their twenties, so if you're younger than that then don't worry too much about vocal classification right now.
With training could I be a bass?
Your voice is a certain type, and that's not something anybody can change. A bass who develops a ton of head voice to hit high Cs doesn't turn into a tenor, he's still a bass with a great high range. Voice type is about where your voice is naturally most comfortable, not absolute ranges.
I heard it is only possible to extend upwards, but I'm not sure how true that is.
It's mostly true. To sing lower you relax the muscles that support your vocal folds, and you can't relax any more than "fully relaxed". Since not using a muscle isn't very hard, most people start out with access to their lowest possible notes. Some beginners unknowingly add tension and can unlock an extra half step or two as they learn to fully relax, but you're unlikely to develop much more than that.
Also how do I know exactly when I go into falsetto? I can't put my finger on my highest note as I don't know which notes count/when I'm "cheating".
You sound a lot like me when I was untrained. I'm guessing you don't have any head voice, so you've learned to mix falsetto in order to hit the pitches that you hear other people singing. It's difficult, and even dangerous, to try to suggest exercises without being there in person to see what's going on, so I would strongly recommend working with a good voice teacher, even if for just a few lessons.