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In videos (1) and (2), we see that the supply and demand curves do not touch the axes.
In (3), we see that the demand curve touches the Y-axis but doesn't touch the X-axis. MR curve touches both axes. On the other hand, MC and ATC curves do not touch any axis.
When do the curves touch the axes and when don't they?
Lets first try to understand what it means: when a demand / supply curves touch the axes.
The point where the demand curve touches the Y-axis (Price-axis) can be interpreted as the price which makes the first consumer willing to pay for that good (prohibitive price). The point where the demand curve touches the X-axis (Quantity-axis) can be interpreted as how many people would consume this good if the price is zero (Saturation quantity).
The question whether every demand curve should have this points translates than to: Could you imagine a price for the good that is so high, so that nobody wants to buy it anymore?
Example. 1 Mio for a Donut.
And could you imagine a quantity where consumer stop to consume even if prices are zero?
Example: 10 free Donuts per day.
Regarding the Marginal and Average Total Cost Curves, they do not touch the x-Axis in a normal framework which I assume are the relevant ones for you. If they do it would imply that in some part of the production the next produced good would have zero costs (MC) or average costs are zero (ATC).
If you are interested in the question why some demand functions are linear and some convex you should read about concave utility functions and their implications.
Broadly, the answer to your question is it depends on the context. Generally, if you have some sort of functional form for the curves, you can tell whether they touch the axes by seeing if there is an intercept on either the P or Q axis (set P = 0 to see if there is a Q-intercept, and vice versa).
So, for example, if you're working on a monopoly problem that assumes a linear demand function of the form $Q_d = a - bP$, the demand curve will intersect the P-axis at $P = a/b$ and the Q-axis at $Q = a$.
It's pretty standard to assume that Q is never negative, so you'll generally never see any curve go to the left of the P-axis. For some curves (like supply), it doesn't make sense for the curve to exist when P is negative, so those won't go below the Q-axis, while for others (like MR), that can still make sense.
But, if you're just doing a general analysis to see what happens conceptually when, say, demand shifts to the right, or supply shifts to the left, then it's irrelevant whether the curves intersect the axes or not. If you don't know that a curve crosses an axis (like from the functional form), it's usually best not to assume either: don't assume it crosses, but don't assume it doesn't cross.
