Show me how to do it

... because at the time she told me so, she WAS going to go.
(Her actual words, back then, were presumably: 'I am going ...' ~ but it was in the past that she said so.)

' ... Has been rising ... ' is better than 'Have ... ' because 'number' (itself) is singular. Using the Present Tense helps to suggest that the rise is continuing even now (perhaps only a few hours or days after the report). If we had said ' ... HAD been rising', it would suggest that we understood what he told us, but we believed the rise had now stopped ~ hence the use of the Past Perfect (or 'Pluperfect'), expressing action two steps back into the past (1 step for the report, another step for the facts that went into it beforehand).

Again (cf. Q.2 above), the Pluperfect is best here, since the research is one step further back into the past than the government's later response.

The verb should be past-continuous, since it probably took him longer to smoke the cigar than the photograph took to be taken. (There was much discussion in Brunel's 150th anniversary year, about whether or not modern reproductions of this famous picture should have the cigar 'edited out' ~ as it set a bad example to children, somehow suggesting that smoking and success were linked!)

'Without his help I would not have known ...' (or even 'could not'!).
In English we know 'what to do' with a machine or gadget; not 'how to make', or any other such version.
(We English are sometimes surprised at how many other languages use one same word for 'doing' and 'making' [French: faire; German: machen] when these are, to us, two quite distinct ideas. We 'do some cooking' [= an activity] but 'make a cake' [= a tangible product]. Come to that, we have separate words for 'wearing' and 'carrying' [Fr: porter; Ger.: tragen] which aren't quite the same thing either: you might take off a raincoat and hang it over your arm or shoulder, if the rain stops and the weather turns warm and perhaps rather humid. On the other hand, these two and many other languages distinguish between different types of 'knowing' ... knowing things we've deliberately, schematically learned [such as languages, sports and how to play musical instruments], and 'just knowing' people and places that we've come up against as we make our way through life. Perhaps that is a distinction worth drawing!)

Warnings are also given in English 'against doing something' ~ or, more simply as here, 'not to do' it.

Answer 1 was wrong because 'people' are plural; 2 is wrong because the Present Tense may have applied during the evening itself, but the event is now over so we need a Past form; similarly with Answer 3.

Good little old 'to' is all we need here. There does not need to be any further future verb element, even though the assembly of the thing hasn't yet begun and therefore does lie in the future.

Presumably 'this article' is more or less current, and 2023 remains in the future, so 'are going to' fits best here.

This is the use of 'could' in a rather exaggerated sense, presumably meaning that the young man wanted the Doctor to put it somewhere fairly unprintable (probably inside his [= the Doctor's] own body) because the young man disliked, distrusted or despised it in some way ~ and felt that the most appropriate way of dealing with this would be for the Doctor to 'have a taste of his own medicine' (as the old phrase puts it).
The 'could ' is obviously theoretical ~ as the Doctor probably had no intention of applying the 'invention' (whatever it actually was!) to himself; or at least, not in quite such a way as the sentence suggests. The young man's actual words were presumably along such lines as 'Go away, Doctor, and you can take your invention with it and stick it ... [wherever] ...'; the 'could' in Answer 2 is the reported version of that 'can'.
More than enough on this, we feel!