일어서다 is a bit more precise word than 일어나다, and similarily "to stand up" is more precise than "to get up". For example, 일어나다 can also mean "to leave bed" or "to sit up", while 일어서다 does not seem to convey these meanings. Anyways, I am not saying that you are wrong, but I would advise some caution regarding the subtle difference in meaning here.
Yes. But worth noting that when speaking of people's posture, the 2 expressions are not always synonymous.
일어나다 describes a change of posture which could be from lying down to sitting up or from sitting up to being up on your feet.
일어서다 is to stand up i.e. get up from one's sitting position.
So, 일어나다 can be used to mean 일어서다, if used to describe the change in posture from sitting to standing. (as you have rightly pointed out. )
But the reverse is not always true.
When using the plain speech style, (으)라 is added to the verb stem in order to make a command. If you use 하다, the verb stem is 하, and since it ends in a vowel, you attach 라 to get 하라, where you're commanding that someone do something. If you use 먹다, to eat, your verb stem is 먹, and as this ends in a consonant, you add 으라 to get 먹으라, where you're commanding someone to eat. So here, where you use the verb 일어서다, to stand up, you take the verb stem, 일어서, add on 라, and you get 일어서라, where you're ordering someone to stand up.