Can a non-citizen vote? Own property? Work?
If they can't own property or work, then how can they live in the country long enough to obtain citizenship?
If they can work, what is stopping a massive migration of individuals from other countries to move here for work?
If they can own property, what is stopping foreigners to buy property without actually entering the country?
I would say that a non-citizen can work.
For purposes of owning property I'd see two separate ways to go:
1) Anyone can own property assuming they pay any relevant taxes, i.e. factories could be taxed on their output, stores on their sales, houses with property tax, etc.
2) Non-citizens are allowed to own property assuming they are actively living within the country and working towards becoming citizens (which basically just means living there and not committing crimes). If a non-citizen fails to comply in this requirement then any owned property immediately defaults to the government.
For purposes of a massive migration, I'm not sure what is supposed to be bad about that. It would probably cause some crazy economics at the beginning, but once it settled down a bit you would have a powerhouse of a work base to go from (and since everyone would be a legal immigrant they should all be able to pay taxes). You would probably have some bad problems with unemployment at the beginning, but eventually the large amounts of cheap labor should draw in more jobs and it should stabilize out.
For voting I would probably say that only citizens have the right to vote.
As for re-election things, keep in mind that the more people you go through the less likely the system will remain stable. As a metaphor imagine a very large library. Now at this library the head librarian has a particular way of sorting books that he likes, and so he orders everyone to start using this method. As a result the shelvers jump to it, but before they are even close to being done with the massive task the head librarian is transferred to a different library and a new one comes in. This new head librarian has his own particular way of sorting books that he likes, and so he orders everyone to start using that method, and so one. The end result is that you end up with a bunch of eclectic systems, none of which are complete, and none of which work together fairly well.
My suggestion is that you double or triple the number of slots for each position of power (so you would have two presidents, for example), increase the term length, and then stagger the elections. (So instead of electing two new presidents with their own ideas of things every 4 years, you would elect one president every 3 years, with them serving 6 year terms, or 1 every 4 years with them serving 8 year terms). You could keep the restriction about not being able to run for re-election. That would allow your system to still constantly change as your voter base changed, but still maintain a sense of continuity and stability since you will always have at least one person in power from "the old way".