"I am standing at the hospital entrance."
We havent learned the kanji you used yet, so for those unfamiliar, 待つ is "to wait".
So he's using で with the verb "to wait", and に with the the verb "to stand", but I'm too much of a noob to know why the particle would be different for either verb.
In my English head they are both "at" the place, so I would figure it is で in both cases... ¯_(ツ)_/¯
From what I've heard, it seems that both に and で are used for places, but there's a nuance depending on the verb used:
に is used for places where the verb used implies nothing more than being in this place. 台所（だいどころ）にいます - I am in the kitchen.
で is used for places where the verb used implies that the subject does something. 台所で食べます - I eat in the kitchen.
So, if I understand correctly, びょういんの入り口でたっています。should use に instead of で because the verb たつ in this form たっています implies that you don't stand up, you are already standing, doing nothing.
That should mean that in KMosuzu exemple, when the verb is 待つ (to wait), we should use で instead of に because it seems that "waiting" is considered as "doing something".
I'm not 100% sure, correct me if I'm wrong ! (Also, I'm french so... sorry for my english)
I also want to point this out (given your comment was written 7 months ago) your English is absolutely perfect with no grammatical mistakes in this comment, don't worry ^_^
the only error i saw was the spelling of "example" and you should say 's for "の”
so it should be "KMosuzu's example" ^_^
The て-form can be used for commands, but it has many uses beyond that.
If you were to use 立つ／立ちます then you would effectively be saying "I stand at the hospital entrance" or "I will stand at the hospital entrance". Which is to say that it is either something that you do habitually or something that you are planning on doing. Either way, it does not indicate that you are doing it right now, as the English sentence does.
To indicate this in Japanese you conjugate the verb into its て-form and then add いる／います.
静置 is not a normal word that people would use. It can't be used in this sentence. It kind of means "to put" or "let stand" so it isn't used for people.
You would use it for sentences like "set it aside for 20 minutes" or "leave to stand for 1 hour."
It is more of a technical word used for a process.