Let's start out by parsing your good sentences, #1.b, #2.b, #2.c, by using brackets to identity the internal complement(s) for the verb "shows":
1.b) The table shows [the average amount of time(i) (that) advertisements on the Internet last __(i) ].
2.b) The graph shows [what proportion of UK adolescents follow a vegetarian diet].
2.c) The graph shows [the proportion of UK adolescents following a vegetarian diet].
You can see from the above that there seems to be three different types of constructions involved, w.r.t. show's internal complement(s). Each one would probably involve a good hunk of time and text to explain. I'm not sure if you're really interested in such a detailed grammatical explanation. If you actually are, then you might want to pose a new question where you asked about exactly one of those example sentences (perhaps it could be paired with one of the ungrammatical examples).
Also, you might notice something if you parse the ungrammatical examples and then compare them to the good ones. But, unfortunately, that might need to have the good ones grammatically explained first.
Anyway, let me give some quick impressions w.r.t. your first example set.
- 1.b) The table shows [the average amount of time(i) (that) advertisements on the Internet last __(i) ].
The expression that is in italics "(that) advertisements on the Internet last __(i)" seems to be a relative clause, which modifies the nominal within the noun phrase "the average amount of time". The relativized gap ("_(i)") is linked to that same nominal. A possible interpretation for that relative clause could be something like: advertisements on the Internet last [an average of five hours].
You can try to use that explanation as you attempt to parse the ungrammatical version #1. Maybe that explanation will be helpful, maybe it won't. (Note that, in general, relative clauses are finite clauses. But the complement in version #1 seems to involve a non-finite clause as a modifier.)
And then, you can attempt to create explanations for your other good versions, #2.b and #2.c.
Hope some of this will be helpful to you.