The Corpus for NOW
Data that may help in discovering the answer to your question were recently released at corpus.byu.edu. Among other corpora, the NOW Corpus (News on the Web), with data from 2010-2016, can be used to analyze use of 'hysterical' in the sense of "funny" in the overall middling-formal English commonly found in online newspapers and journals.
The newspapers supplying text for the NOW corpus come from a variety of countries.
- How common is the use of the term hysterical, meaning “funny”, in BrE and AmE?
To narrow the scope of this question, I looked at the context of instances of 'hysterical' in the NOW corpus with an eye to the relative frequency of occurrence of 'hysterical' used in the sense of "funny" compared to instances of 'hysterical' used in other senses. Because 'hysterical' occurred 5419 times (compare that number to the 158 occurrences of 'hysterically funny') in the corpus, I was compelled to sample the occurrences, rather than treat them exhaustively.
The 5419 instances of 'hysterical' in "about three billion words of data from web-based newspapers from 2010 to the present time" were predominantly used with other senses than 'funny'. A semantic breakdown of the results is not (yet) automated, so I scanned and evaluated samples of the use in context. In the 130 instances (some of them duplicates) of 'hysterical' found May 1-19 of 2016, for example, I observed only 10 uses that I could positively identify as meaning "funny". Extrapolating from that to the total, approximately 400 out of 5419 occurrences of 'hysterical' in the corpus are used in the sense of "funny". I see no reason to suppose that exhaustive examination of the 5419 instances from 2010-2016 would find an overall greater or lesser proportion of 'hysterical' in the sense of "funny", but considering that I only examined a very small sample (130 out of 5419), it is certainly possible that 'hysterical' in the sense of "funny" is more or less common than my finding suggests.
- Does 'hysterical' actually carry a negative connotation as suggested in the above extract, or does it only convey a neutral meaning?
Connotations are, of course, completely context-dependent. My observations of the use of 'hysterical' in the sense of "funny" in context for the 130 instances I examined were that 'hysterical' was primarily used postively, that is, to laud whatever was called 'hysterical' in the sense of "funny". Other senses than "funny", however, were predominantly negative, but were sometimes neutral. I did not observe a use of 'hysterical', in a sense other than "funny", that was positive in context.
- Is its usage currently on the downtrend?
This is much more difficult to determine, and the determination more difficult to support authoritatively: the analysis requires comparisons of samples, in context, month-to-month, year-to-year.
A rapid scan of the last 100 occurrences in April turned up approximately the same proportion of use in the sense of "funny" as I found in the 130 instances May 1-19. In accordance with that rather scanty and unsure correspondence, I ran a frequency distribution analysis:

[Discrepancy in data: The total freqency for all periods in this chart is 5823, rather than the 5419 presented as the frequency for a simple list. Subtracting the suspect duplicate of 449 in periods 2012-B and 2013-A gives an overall frequency of 5374, which also does not match 5419. I have no ready explanation for the discrepancy.]
As can be seen from the "PER MIL" chart column, a downward trend for use of 'hysterical' in all senses is evident year-over-year in this corpus. If my conjecture based on small, recent sample sizes is accurate over the timespan of the data, that downward trend would be reflected in a corresponding proportional downward trend in the use of 'hysterical' in the sense of "funny".
Notes
However fond the men and women of the press are of seeing themselves as vox populi, that description is not true to what is now the intrinsic purpose of their endeavors, selling or at least circulating 'news'. They use what can only be called a dialect of English, and the topics entertained by their work are restricted by considerations not foremost in the minds of many, nay, most speakers: that is, the motivating force for commercial 'news' is commercial gain, along with satisfying the whims as well as the social, political, and other predilections of the owners of the mass media.
This being the case, however, leaves the question of what more accurate collection of the uses of language by ordinary people is available as data for our consideration. Some researchers have resorted to sending correspondents abroad to record and transcribe the speech of what it is hoped will be a representative sample of the speech of various populations, and the results of those collections have been made available. The extremely small sample size, however, leaves any question with respect to a particular word, such as 'hysterical', in doubt.
Other Corpora
Another useful corpus from the same source (corpus.byu.edu) may be the Corpus of Global Web-Based English (GloWbE) (pronounced 'globe'). This has drawbacks:
- It is a snapshot corpus compiled in 2012, so it doesn't include the four years of language and web development after that time.
- GloWbe has, of necessity, more 'noise' in the data. The 'noise' may consist, for example, of more duplicated text, or of simply random text. The researchers who compiled the corpus made a decent effort to remove duplicated text in the form of boilerplate material, for example, but the automated techniques necessary to find and remove such materials are...less than perfect.
- Expectably, GloWbe instances cannot be dated accurately.
Nonetheless, I made a brief sortie at the dataset, which "is composed of 1.9 billion words in 1.8 million web pages from 20 different English-speaking countries. The web pages were collected in December 2012". See the description of the method used to create the corpus for more information.

The occurrence of 'hysterical' 5649 times in two-thirds the number of words as comprise the NOW corpus may suggest that in a broader range of less formal English, 'hysterical' is used more often. As with the NOW corpus, I analyzed approximately two hundred occurrences in context (60 US, 60 GB, 40 CA, 40 IE) to see what proportion of the uses were in the sense of "funny".
Of the 60 US, 11 were used in the sense of "funny"; of the 60 GB, 11; of the 40 CA, 9; of the 40 IE, 5. US = US English; GB = British English; CA = Canadian English; IE = Irish English.
Proportionally, then, the frequency of use of 'hysterical' in the sense of "funny" is higher, 36 out of 200, in this corpus than in NOW. The predominant use, however, remains with other senses than "funny".
More or less just because it was there, I also checked in the Corpus of American Soap Operas (SOAP). This corpus "contains 100 million words of data from 22,000 transcripts from American soap operas from the early 2000s, and it serves as a great resource to look at very informal language."
'Hysterical' occurred in the SOAP corpus 474 times overall, 4.74 times per million words, a higher frequency than any section of the NOW corpus. Of the 100 occurrences in context that I checked, 19 of them were in the sense of "funny", and 5 of those were sarcastic uses in that sense.
These observations suggest that in "very informal" English, both the overall frequency of use of 'hysterical' and the proportional frequency of use in the sense of "funny" are higher. However, both the overall greater frequency and the greater frequency of particular use may be artifacts resulting from the different time periods documented: NOW is compiled from data gathered 2010-present; SOAP was compiled from data extant in "the early 2000s", 2001-2012 (see chart for frequency per million per year); GloWbe was a snapshot of data in 2012.

In addition to the frequency differences, sarcastic use of 'hysterical' in the sense of "funny" was conspicuous in this corpus, but rare to the point of vanishing in NOW and GloWbe.
Also of interest in the SOAP corpus is that 'funny' is one of the top ten collocates, the tenth most common:

A quick look at those collocations in snippets of context shows that all are uses of 'hysterical' in the sense of "funny", although whether or not the use is sarcastic remains opaque without seeing more context:

Nonetheless, personal experience (also known as anecdotal evidence) aside, the use of 'hysterical' in other senses than "funny" was predominant in this as well as in the other corpora examined.