I don't see why the position of the President cannot be a simple one.
1. Give him the power to veto
2. Give him the power to ask the House to dismiss the government/prime minister in a vote of no confidence.
3. Give him the power to introduce legilslation to the House in the form of a 'Presidential Slot' if you will where he has the power to introduce one bill at a time.
This makes him exercise power and also push for a legislative programme of sorts, which makes a campaign and a nationwide vote worthwhile.
I agree with this. This way, the President can check the PM if he gets out of control.
In a way. The President is accountable to the people. The PM is accountable to parliament and is chosen by them - the President can ensure parliament 'play nice' and if it's the President who's acting up he can be removed at the next election by the people.
By "dismiss the government," do you mean call for impeachement of one particular parliamentary offender, or dismiss the entire Parliament? Also, will the President have the power to dismiss a Parliament member without a trial? I think that offenders should be put to trial at the suggestion of the President. I don't think the President should have the power to arbitrarily choose whether the Parliament is acting properly. I think he should only have the power to ask for a trial.
I also note that there is no mention of the Supreme Court. I believe we should have a three member Supreme Court, appointed by the Prime Minister. Also, impeachment trials should be held in front of the Supreme Court. This way the President does not have the Supreme Court in the palm of his hands, but neither does the PM, because only the President can call for impeachment.