You can think of て as having 3 basic uses.
The first is to show sequence. A happens, then B, then C. ご飯を食べて、シャワーを浴びて、寝ました。I ate my meal, then took a shower and then slept.
The second is to show means, method. A is used to do B. この机は、私が木を使って作りました。As for this table, I made it by using wood.
The third is to show reason. 地震があって、大勢の人が死にました。Because an earthquake happened, lots of people died.
Perhaps you could now think about how each of your two examples fits in with one of the three usages of て.
For me, I think of it this way:
離れて見る can be thought of as the second usage: Through/by being at a distance, I (or some other person) look(s) or see(s) something. The method of seeing or viewing - the "how" - is by being far away.
事故があって、バスが遅れてしまいました。can be thought of as the third usage: The bus was late because an accident happened. (note the nuance of regret or disappointment that I can't really incorporate in the translation. This is because of てしまいました.) て in this case shows reason.
Note: the third usage of て, to show reason, is limited by at least two rules. When you construct a sentence with "reason て result", what follows the て must be non-volitional (cannot be controlled / happens naturally). Furthermore, the reason must not happen after the result.
For example (edited with chocolate's suggested sentences):
O お腹が空いているので/から、ケーキを二つ食べましょう。(ので and から can take volition)
O お腹が空いていて、ケーキを二つ食べ(てしまい)ました。/食べられました。(て with non-volition)
X 友達があした来て、きょうはいそがしくなりました。 Because my friend is coming tomorrow, today became busy (became a busy day). My friend coming tomorrow happens, chronologically speaking, after today became busy. Thus the sentence becomes ungrammatical.
O 友達があした来ますから、きょうはいそがしくなりました。Because my friend is coming tomorrow, today became busy (became a busy day). から is able to express a reason happening after a result, so this sentence is acceptable.