1>. Why are candidates (and their reps) allowed to even see any votes before they have all been cast?
Technically the postal votes they saw already had been cast. Candidates/agents get to see/monitor the counting of all votes to enable the process to be considered fair - so if postal votes start to be counted early then they need to be able to observe this.
2>. Why are postal votes even opened before the ballot is complete?
Becuase in the UK we still have this idea that we want the election to be determined as soon as possible - we expect to know the new PM on the next day and this time having to wait 4 or 5 days was seen to be quite a problem - there were real concerns over how markets might react when there was no decision by monday morning.
3>. Why do postal votes even exist if we cant verify the eligibility of the voter?
Postal votes have always existed and the verification techniques now, I think, are a lot more robust than they used to be (e.g. a signature that can be matched to the application is required). They are much more widespread now because it was seen as a way to get many more people voting. N.b. "in person" votes used to be pretty insecure - just turn up at 8am at a poling station with a name and address from the electoral roll and you'd get a ballot with no need to prove identity - not sure if its still the same as I've voted by post for the last several years!
Again, because we want the result "quickly" we need to have a system that allows postal votes to be "verified" quickly ... there is provision to challenge a result later in the courts but that only overturns a result at a later date.
The alternative is exemplified by an exchange during the 2000 US election between a BBC reporter and a UK representative of one of the US parties. As the result became so close in Florida (and before the "hanging chad" issue had been raised) the US person commented that the result might have to wait until the "absentee ballots" were certified - the BBC person asked how long that would take clearly expecting an answer along the lines of "by lunchtime" and was stunned to be told "in 21 days time" ... because under Florida election law the parties had 21 days to challenge any postal votes and the result was not finalised until that time was over - normally the number of postal votes would be less than the victory margin so it wasn't an issue but in this case it was (and, I think , this also contributed to the courts ruling out the hanging chad debate on the basis that a decision had to be ratified after 21 days).