(1) Happen (Cause)
Yes, you are right.
Both "起こる（おこる）" and "起きる（おきる）" have the meaning of "happen."
○「問題 が 起こる（もんだい が おこる）」
○「問題 が 起きる（もんだい が おきる）」
both means "Problem happens."
○「ストレス で 起こる 病気」（ストレス で おこる びょうき）
○「ストレス で 起きる 病気」（ストレス で おきる びょうき）
both means "Illness caused by stress."
"起こる" is more traditional word in Japanese to say "(something) happen(s),"
but recently, more and more people use "起きる" to describe the same situation.
Both are still very commonly used phrases in Japanese,
so if you are trying to say “(something) happens,” most of the time these two words are replaceable.
(2) Get up = 起きる ≠ 起こる
However, "起きる" also commonly used in the meaning of “(someone) get(s) up.”
○「朝 早く 起きる（あさ はやく おきる）」
means “wake up early in the morning."
In this case, since "起こる" does not have the same meaning, you cannot say
×「朝 早く 起こる（あさ はやく おこる）」
(if you say so, it means "it happens early in the morning.")