Global property does not imply local property in topology

Question

I just learned that path connected spaces are not necessarily locally path-connected. The classic counterexample is the Comb space.

Question 1: Is this the main issue:

Let $X$ be path connected. Let $x\in X$ and let $U$ be a neighborhood of $X$. Let $a$ and $b$ be in any neighborhood $V\subset U$ also containing $x$. Since $a,b\in V\subset U\subset X$, there is a path in $X$ from $a$ to $b$. But it does not necessarily lie entirely in $V$, and this is what the counterexamples target, right?

Question 2: It is really strange to me that a property could hold globally but not locally, so in addition I must ask: are there other examples of this? Because doesn't, e.g. "bounded $\implies$ locally bounded"?

From googling, it seems localizing these topological properties is not exactly standardized yet:

1. The answer here shows every compact space is locally compact
2. The answer here finds a space that is (quasi) compact but not locally compact

This seems a little sloppy to me--that the same name is given to two different notions of "locally property $P$". Munkres' text goes with the definition that makes compact spaces automatically locally compact and does not seem to mention the other one.

If anyone could provide some references, intuition, on why this terminology makes sense when some properties "global $\implies$ local" and others do not, it would be greatly appreciated.