When do the curves touch the axes and when don't they? The Demand Curve
The Supply Curve
Monopoly: How to Graph It

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?
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.