+0
In brackets is my presumption

1.She just finished her test and the bell rang.
Hardly (had she finished her test when the bell rang).
2. I have hardly ever seen him dance.
Selodm (have I seen him dance)
3. No matter what happens don't let this out to anyone.
Under no circumstances ( lets this out to anyone.)
4. They could not figure out how they could cope with so much work in so short a time.
In no way ( could they figure out how they could cope with so much work in so short a time.
5. The moment she was through with the ironing she started washing the dishes.
No sooner (was she through with the ironing than she started washing the dishes).
6. Everybody thinks much of her.
All ( are thought much of her) maybe it is wrong.
7. Must be going or else i shall be late.
It is ( I have no idea )
8. The first time I heard of him was when his first novel came out.
Only (when his first novel came out did I hear of him)
9. I asked not Jane but Gill to help me.
It (............) I have no idea of the way of construction.
10. " I wonder if my nephew will come to my party tomorrow" she kept thinking.
She kept wondering ( if her nephew would come to her party next day)
+0
1,2-- OK
3-- 'let'
4, 5-- OK
6-- wrong indeed. 'All think...' Try again.
7-- 'It is late, so...' Try again'
8-- OK
9-- 'It was Jane not Gill...' Try again
10- OK, though I would use 'whether'.
Comments  
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6 - All think much of her. It seems to easy to be in this way!
7 - It is late so I must go
9 - It was Jane not Gill asked to help me .

I have some questions.
The form " must be verb+ing" could you explain to me the usage of this form. When we use it because I have some grammar books but non of them have such a section.
Second why in the 3 sentence there is no inversion! I know that after under no circustances always has an inversion! I suppose that it our situation we don't make an inversion because we have "let" ! and from that I concluded that with let we do not make inversion. My presumptions are right or wrong ?
6-- Easy, but I think that must be the answer
7-- I think I made a mistake with my hint, Max. I think rather 'It is getting late, so I must go'. Still, that doesn't carry the same meaning. I don't know whether I can't think of the answer, or the question writer was sloppy.
9-- Almost; and I made another mistake with my hint! 'It was Gill not Jane whom I asked to help me'.

'Must be + going' here means that outside powers are forcing me to leave, though I would much rather stay. It is a form of courtesy. The use of the continuous reinforces its immediacy for the speaker.

I suppose because it is imperative and has no overt subject. If we rephrase it, it would invert: 'Under no cirumstances should you let this out to anyone'.
From your answers I understand everything but one !
Why did you use whom not who ? In
It was Jill not Jane whom I asked to help me.
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Both 'who' and 'whom' are correct and in use, but 'whom' as the object of 'ask' is more formally accepted, and is probably the answer sought by the writer of your exercise.