This is all very confusing to me also being a Hindi beginner. But I've just checked the oxford Hindi-English dictionary and I think I have an answer. There is no verb "to stand" in the form of "खड़ना" (this would be the basic form of the verb called: infinitive). That's exactly what it says in the exercise when you point the mouse on the word - it shows translation and explanation. The word "खड़ा" is an adjective, so you can't say "... खड़ / खड़ा / खड़ी रही है". It's not a verb so it doesn't conjugate! You can use it for example: "वह खड़ी लड़की मेरी बहन हैै." I think that in this sentence the authors want to show us that there is another way of building present continuous tense in Hindi: by using adjectives. This structure is somewhat similar to English "Can you see that standing girl?" but there is no exact equivalent in English grammar. But I have a very similar structure in my native toungue, that's why I think I'm right. So this adjective only takes the masculine or feminine ending depending on the subject, but you don't use it with the verb "रहना", and the parts of the proces of standing have nothing to do with it. I hope this wasn't too confusing. If anyone has other (or "the correct" :) explanation, I would love to know.
Standing, as you allude to, is an adjective in English, as well as being the present continuous form of 'to stand'.
I've commented a few times that 'standing-up' and 'sitting-down' are (commonplace) English phrasal adjectives that would make it clearer that the Hindi translations are not verbs.
('While you are standing up' means 'while you are on your feet', i.e. at full height, not 'while you are rising', i.e. in the process of standing from a sedantry position.)
What you wrote isn't a valid Hindi sentence.
I guess you meant - वह लड़की कुर्सी पर खड़ी हो रही है।
Although this is a valid Hindi sentence, it will be rarely used because of its weird interpretation. It can't be translated in English without additional comment. It would translate to - "That girl is (in the process of )standing on the chair." And this हिंदी sentence forces you to imagine as the girl stands in the chair in slow motion. That's why its an unusual sentence with limited use case.
I think what Predeep means is that the original Hindi sentence is lacking the continuous tense, which would be (I believe): वह लड़की कुर्सी पर खड़ रही है
The translation should be: The girl stands on the chair.
Edit: or am I missing something? I'm no expert by any means, but it's a bit annoying to get things right/wrong randomly. Sometimes continuous tense, sometimes not.
I may not be 100% right. But, What I understand is, in Hindi, present continuous is used when the process continues. For example Sitting,standing and other equivalent verbs means the process or the action of changing from standing to sitting position and the process ends once the person is seated or stood on the chair and in that case we have to use present perfect (बठा). Whereas in case of the verbs like playing, crying,etc the process or action continues till it ends and so we use present continuous.
Ex: वह बैठा है - He is sitting (Already seated)
वह बैठ रहा है - He is in the process of going from standing to sitting position
वह बैठता है - He sits (Simple present)
Here, just the रह is eliminated and | is added to बठ to form बठा.