1. ‘Why do we say ‘I’m broke’? Why not ‘I’m broken’?

The three forms of break are: break, broke, broken. Broke has the meaning as in The glass pane broke. Seeing such a window you say: ‘The glass is broken’

But the past form broke has also a figurative sense in: ‘I’m flat broke’ (having no money, penniless, bankrupt). Broke in this usage, is purely adjectival.

Broken also can be used in an adjectival sense. The death of his wife devastated him. He is now a broken man. The meaning here is ’weak, exhausted, dispirited and melancholy’.

2. The vegetables heap into a mountain. Is it possible to have a passive on this?

No. There is no object here to become the subject of a passive sentence. ‘Into a mountain’ is an adverbial phrase.

As for The vegetables got heaped into a mountain, this is what is called the get-passive.  You can supply an understood agent: got heaped into a mountain by the vendor or some such thing.

3. Can I ever play? Can I never play? What’s the difference? Are they synonymous?

The first sentence expresses a remote possibility. Thus a tennis player with a surgically treated broken wrist may ask: ‘Can I ever play tennis again?’ (There is a remote possibility that he may be able to play again.)  But where the chances are practically nil, he would ask: ‘Can I never play again?’

4. The end /in the end.

‘The end’ is the farthest point of something which has extension: the end of the road/ the end of the journey / the end of the world  (in the last case, extension in time).

‘In the end’ refers to the last phase of some activity. He tried many jobs; in the end he became a postman (ALD).
As you can see the two phrases don’t mean the same. ‘In the end’ presupposes some earlier activity.

5. Kaif has been dropped for both the one day as well  as the test series. Do we need both?

No, we don’t. As Fowler remarked, ‘to follow both with as well as instead of and, as is often done either by inadvertence or in pursuit of the unusual is absurd.’  Fowler’s explanation (or elaboration) of his point cannot be quoted here as it is somewhat technical and calls for further explanation by me. I shall merely point out that both…as well as is a tautology, a fault of style. Replace as well as by and, if you want to keep both, or just drop both  if you want to keep as well as;  from both the one day and the test series / from the one day as well as the test series.

Fowler points out a further fault in the use of both which is not exactly a tautology.  If any great advance is to be at once both intelligible and interesting…At once…both is not a tautology. The two expressions don’t mean the same.  But at once adds nothing much to the meaning. Dropping it leads to no confusion or loss of meaning. Fowler’s advice is to drop at once. (More on this later.)

6. Al Quaida is reconstituting itself in Northwest Pakistan at a time the very stability of that nuclear armed nation is in question. Quoting this sentence Prof Sajjan asks whether it is correct to omit the connecting word when here.

The connecting word in a sentence containing a relative clause can be omitted when the word (the relative pronoun) is not itself the subject of the clause. The man who you met at the theatre yesterday…. Here the subject of the relative clause is you, not who. So we can drop who. The man you met at the theatre yesterday. 

But in the man who came yesterday is an architect, who is the subject of the relative clause. So you cannot drop it. If you drop it you get the ungrammatical sentence the man came yesterday is an architect. Again, the relative pronoun cannot be omitted if it is in the possessive form. In the man whose brother we met at the theatre, the subject of the relative clause is we, not whose brother. Still we cannot drop whose. If we did we get the ungrammatical structure The man brother we met at the theatre…. The picture is a little more complicated but this will do for now.

Coming to words like when, where, why, etc. (they are called relative adverbs) the restrictions noted above do not apply. A relative adverb cannot be a subject; it cannot be in a possessive form.  So it should be generally possible to omit it. Here is a citation. From the time he left Harrow to the present day he had a passion for combat. Cf. From the time when he left Harrow…(G. Scheurweghs: Present-Day English Syntax,1959).

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