Firstly, these are not petitions to Parliament but petitions to government. The government doesn't dictate what MPs debate in the Commons; there is a delicate arrangement around government business that is managed by the Leader of the House and Lord Privy Seal, currently Sir George Young, who is both the government's representative in the Commons and the Commons' representative in government. Any debate resulting from the e-Petitions site would presumably have to be introduced as government business amidst an already crowded and pressured agenda of legislative matters. The fact that this initiative hasn't been introduced by the Speaker but by the Cabinet Office speaks volumes. Speaker Bercow could redeem the entire car-crash of his tenure in one move by enabling e-Petitions to Parliament, but this is too close to direct democracy to be welcome by our political class.
Secondly, any petition must be approved in advance by the senior mandarins of a specific government department - presumably either the Home Office or Ministry of Justice for the death penalty, and the Cabinet Office or Foreign Office for EU matters. There will no doubt be a screening process under which petitions for measures contrary to law, such as the Human Rights Act or Lisbon
Thirdly, as the experiment on the No 10 petitions site proved, allowing numerous petitions that are variations on a theme is a good way of guaranteeing that no single one gains the necessary minimum vote, so petitions to "re-introduce the death penalty for terrorism", "bring back hanging for paedophiles", "introduce capital punishment for treason' and 'bring back hanging' will cause public confusion and scattergun votes.
Fourthly, the measures contain no provision to secure a referendum on any issue, no matter how strongly supported. The issue will at best go to earth in the Commons. Helena Kennedy's 'Power' Commission came up with a workable model for this kind of direct democracy which has been rejected out-of-hand by the government; Kennedy's proposals were
1. A trigger of 400,000 votes to guarantee a debate and vote in the Commons with no censorship by government
2. A further 400,000 votes to secure a referendum if the Commons have rejected, modified or diluted the original proposal
3. A simple majority and a turnout of over 60% and the measure passes into law
I really do hope that the British public see through the government's transparent intention to defuse some of the public's anger against a remote and unaccountable political class with a pointless and tokenistic exercise, and boycott the entire thing. But no doubt come the 4th August all the usual issues will be there, a soporific tranquiliser and a weak simulacrum of direct democracy.