1) I am curious about the idea, how would you implement that? If you do not like a candidate, you do not vote for it, the current system is simple enough. Are you suggesting elections to "approve" or "reject" the candidates, and then the actual elections?
2) If they have hidden something that makes them uneligible and this surfaces during the campaign, yes. If no, not. They have met all the legal criteria and so they have the right to present for the post, so it is not anyone's else business to prevent them from presenting if they want to.
3) Yes. Someone else could have won the nomination for each party. Someone else still can (AFAIK) present himself as an alternative, independent candidate.
And in most democratic countries the system is quite similar, in that the political parties must internally work as a democracy1
The fact that you or other people don't like either of the candidates does not mean that there is a reason to throw the democratic process off the board because just a few opinions.
As I posted somewhere else, you do not have the right of being offered a candidate that you like well enough to vote for2; if you do not like any then you may either:
support a candidate that you would vote and convice him to run for office.
present yourself for the position.
simply don't vote (or if possible cast a white or null vote).
The answer of this question may give some (USA-centric) additional insight: What happens if a candidate in a US Presidential election becomes the subject of a serious criminal charge?
1 Note that this does not mean that the candidate must be elected directly by supporters/affiliates; in many cases the candidate is elected by the organization of the party. But that organization must be democratically elected.
2 And most of the electoral systems require that if you run for office you must show some measure of popular support (let it be endorsement by political parties or a petition signed by a minimum number of electors), so if there is a candidate we may assume he/she will get some votes.