Strictly speaking, the answer is unequivocally yes. However, that's extremely misleading if you don't know what a "rule" is.
Rules are things that are handed down to you. Some rules are chosen but make sense (like distributive property). Some rules are chosen because we just had to pick something (like x-coordinate points to the right). Some rules are actually just result of other rules (doing the same thing to both sides of an equation is just inherent to what "=" means, since otherwise the two sides wouldn't be equal).
Many things are NOT rules. Like "Solve 2x-7=0 by adding 7, THEN dividing by 2". You could also divide by 2 first, then add 7/2. There are basically no procedures from algebra that you should memorize. But the above *is* a Strategy: "usually solving by reverse order of operations is easiest." There are also Tricks, like recognizing a difference of squares. You learn a list of tools. The understanding you need there is just when to apply them (which usually comes from experience).
It's important to draw distinctions between dumb tricks, interesting rules, and rules that are just there, because then you know where to spend mental resources. To a degree, it's a matter of taste. But most things in math before proofs classes are not deep. There is a reason behind it, but that reason is usually "that's the definition". Or something like "you use the quadratic formula on quadratic equations because that is the solution to quadratic equations" is all the depth of understanding you need. My personal general rule is to try to feel out anything that's visual/spatial, and to just categorize everything else as "that's the definition" or "that's something that works".
So, like everything in life, don't fall into a camp. Whatever situation you're in, do the thing that applies to that situation. If something is deep, learn more; if it's not, don't. There is no magic reason others seemed to be doing better than you. Some people just act like it's fine, some people learned what I said above from an early age and it snowballs, some people memorize a bunch of stuff that isn't rules and do fine in basic algebra but then bomb in college. Overall: learn the fundamentals, take things for what they are, and don't worry too much. If something's not working, talk to people and try something else (like you are doing now).