“There is a dragon in the grounds!”
"in the castle (barn, church, room, cup, etc.)" ("Within the castle walls, midnight silence reigned.") but "on the grounds" or "within the castle grounds" (but in/on the bed, in/on the road seem interchangeable -- I lay in bed (covered up by the sheet), I lay on the bed (on top of the covers) We sat on the bed. ie, used it as a chair. We sat up in bed. Better not go there.... )
You said: Quickly, I drew my wand out of my pocket, and started to make all the ingredients I would be needing to soar through the air towards my desk, using the Summoning Charm.
... I would need soar ...
Otherwise, you are collecting the ingredients so that YOU CAN SOAR TOWARD YOUR DESK.
Thirty minutes later, I was dashing up the narrow stairs, into the Entrance Hall, and then into the grounds, my black robes billowing behind me.
"was dashing" implies the action is done over and over. When it is a one of a kind action, use simple past, "I dashed up the narrow steps, through the Entrance Hall, then onto the grounds, ...
In the corner of my eye, I saw tiny Professor Flitwick unfolding his hands, as though he had been praying.
Out of the corner...
From the corner...
IN places him INSIDE the corner of your eye. Readers will figure out what you mean, but...you want suspense here, not laughter.
The dragon took a step forwards, it giant foot made the ground shiver.
forward, its giant foot making ...
So did we, and we took quite many steps backwards.
"quite a few" or "many" (consider adding "rapid" after either phrase)
(I'd probably use "then" instead of "and" here, but it is not wrong, just lacks the emphasis the situation seems to require.)
But I have to admit, I didn’t feel too big for my boots when I saw that Madam Hooch was almost grilled by a huge flame from the dragon’s mouth.
(too big for my boots = smart alec, overly authoritative) Not quite the right idiom for this spot. Maybe more straight forward -- I felt a lump in my throat/ I shuddered/ I was horrified (or some other appropriate reaction.)
And as if on cue, as Madam Hooch dodged another flame, twelve wizards in dark blue robes came striding towards us, each of them had their wand out.
(Two complete sentences as written... ; between them, or change to "each of them with their wand out", which is a dependent clause and would use a comma, as written...
The Horntail was now staggering around on the lawn, knocking some trees over with her tail, almost trampling on the gamekeeper’s hut, and finally, with an ear-splitting bang, she hit the ground, and at the same time she squashed the gamekeeper’s beautiful garden totally flat.
Generally, when the action shifts to another character, a new paragraph is begun.
I'd also go to simple past, as, again, this is a one-of-a-kind action, not one that will be repeated daily. The Horntail staggered...
(Wordy) better: with an ear-splitting bang, she hit the ground, squashing the gamekeeper’s beautiful garden totally flat.
Madam Hooch, shivering, but smirking, parked her broomstick in mid-air next to us. She was greeted with cheering and a huge applause.
Either "a huge round of applause" or "loud applause" (or "enthusiastic" or some other appropriate adjective).
The twelve dragon experts rushed off towards the dragon.
("toward the dragon")
In the next bit, "toward" again would be preferred. At least in America, if you want your character to appear to be stupid, uneducated, etc. IN SPEECH, he/she would say "towards" and "forwards", as well as other grammatically incorrect phrases.