‘She is in New York right now. She's staying at the Park Hotel’ is fine. It refers to a single, clearly defined situation and action. What are you asking or suggesting with ‘she/stay’ please?
‘She always stays there when she's in New York’ is fine. It refers to all instances of an action; not just to one or several, but all: at least the two or three which would have been needed to establish the pattern, perhaps hundreds over many years but each individually accountable. What are you suggesting with ‘she/always/stay’?
‘She is always staying there….’ is wholly different. It refers not to one or any countable number of specific instances but to her habit as a general idea.
‘… staying there….’ compares to any repeatable, characteristic action such as ’she is always asking people for money’ or ‘… forgetting where she lives’ or ‘… somehow winning the lottery’ but we use ‘she/he/they is/are always…’ to refer to an idea in the abstract, not to any realisation of that idea.
When your textbook said ‘I always do’ and ‘I’m always doing’ were different, what did it say about that difference? Which parts of that didn’t make sense?
Then, ‘Park Hotel’ might be unique if it took the definite article but ‘a Park Hotel’ could be one of many, perhaps several even in that burgh. Where did anyone get the idea that ‘as for proper nouns, we usually aren't to use the definite article for such…’ and what detail went with that? Sorry but it’s backwards. We usually are to use the definite article for such…
Consider Tommy Atkins, a man from London, a city in England, a country in the United Kingdom…
Mr Tommy Atkins refers to a specific person, however he is identified.
A Tommy Atkins refers to any unspecified one of several people who share that name.
The Tommy Atkins refers to a specific person, commonly identified through some sort of celebrity.