Should 'э' be pronounced as 'a' in "cat", or 'e' in "net"? Is its pronunciation context dependent? If so what are the rules for its pronunciation?
'Э' often is used to transmit the sound [æ] (blackpool) or diphthong [ɛə] (Delaware, Blair).
There are some rules for its pronunciation:
'Э' is pronounced in initial position before hard consonants: этот, энциклопедия, эклиптика, эволюция, элементарно.
'Э' after consonants (пэр, мэр, сэр) denotes the hardness of the consonant.
'Е' reads as 'э' after some hard consonants of loanwords: фонетика, отель, кафе, эссе.
'Э' is used to write a few words of ingrained Russian: этот, этакий, это, эк, эвон, эва, эдак, эх, эхе-хе, эй, эхма, эге-ге, эге.
If you mean the British pronunciation of [æ] as in "cat", this is definitely not our 'э'; it's like 'я' between two palatalized consonants. The closest to stressed Russian 'э' among symbols used for English is [ɛ], but [e] is also acceptable. But, on the other side, Russians usually have difficulties to pronounce [æ] correctly here and replace it with a long [ɛ]; this is supported by some American dialects affection and that can confuse native English speakers to find a relation between sounds.
The Wikipedia page declares [ɛ] as the main variant for a stressed 'э'. It's rather true because the relation between [ɛ] and [e] follows common relations between the mid-back vowel row, after unpalatalized consonants, and the front row, after palatalized consonants. The letter "э" is never used to palatalize a consonant before it, so it means [ɛ]. OTOH, "е" can mean both [e] after a palatalized consonant (the most used case), [e̞] after non-palatalized sibilant as "ш", or [ɛ] in loanwords (тест, кашне, проект, etc.); the native words usually have no non-palatalization effect, but it's typical for loanwords.
You should also notice that Russians who haven't got special linguistics education don't understand the difference between [ɛ] and [e], as with most other pairs ([a] vs. [æ], etc.) because the factor which affects vowel allophone selection is palatalization of the previous consonant, but in written the latter is defined by vowel letter selection (а/я, э/е, etc.); this could be confusing if not understood clearly. Traditional school grammar requires a student to sign the vowel as [э] independently of its implementation variant, so e.g. "лето" is transcribed as [л'эта].
So, the short final conclusion is that you can pronounce it like [e] in "net", etc., or better as the first sound in "air" [ɛə], but the main factor is that you shouldn't palatalize ("soften") the preceding consonant and you will be correctly understood when using either of them; and [æ] isn't applicable here. And, BTW, unstressed 'э' is something between [ɨ], [ɪ], [ɛ]... the difference between them is not important in this context.
Here is an article in the wiki about the sound with sound file, denoted in phonetic transcription as [ɛ], which is the ideal Russian 'Э' from a linguistic POV. However, from the POV of a native speaker this sound denoted in phonetic transcription as [æ] is the same. To be precise, this is a part of English that gives native Russians a hard time.