It's hard to answer everything so I'll just focus on a few key aspects.
Wouldn't forcing senate reelection in case of failed impeachment allow the party that controls the house to force that reelection?
Let's say Party A controls the house while Party B controls the Senate and the Presidency. If Party A wants, they would be able to send impeachment to trial without real reason and then by the fact that Senate didn't convict you would get a reelection of Senate.
Way I see it, this would allow manipulation of the election process. In my mind, it would be much more logical that the House has to be reelected as they pushed forward an impeachment that was not convicted upon. This would make it so that House would not bring forward frivolous impeachments as they would have their own positions on the line.
About unseating a president, I understand your point. I agree to a degree. A sitting president can be limited in power by the fact he doesn't control the legislative branch. This may seem less optimal but it has a couple of advantages. Executive branch can keep on working within the confines allowed to them, without having to juggle for political support in the parliament. This allows the President to keep working with little regard to losing his position. This is especially good for military or foreign affairs as they can work on this with little to no political support.
Right, but that's a plus - that's more how it works in other democracies; the equivalent of the house having the upper hand; essentially impeachment at that point turns into a milder form of a vote of no confidence. It's still milder than in other countries, because the senate gets a vote at all.
Framing this as if any of the legal constructs (house, senate, presidency) is reasonable, and thus needs to perhaps be punished for abuse of power is I believe pointless. Impeachment is not about right or wrong, it's about political support (I mean, just look at those almost completely party-line votes in house and senate recently!)
For American democracy to start working again - which has nothing to do with fairness, mind you - the power hierarchy needs to clearly weed out obstructionism; and where obstructionism is possible or even valuable as a kind of common sense sanity-check, either side involved needs to be elected or appointed on the same electoral basis (because the other option is political powerplay, rendering the whole point of things like impeachment moot, as it is today).
Pick two: A government that can govern, or elections with differing structures (essentially ensuring various branches will fight simply as a form of party politics, not based on merit), or equal branches of government, where one (plain majority) cannot remove the other.
You can't have all three.
Aside, the idea that the presidency is stable because it's decoupled from congressional support is not born out by simple comparison to other countries. Quite the opposite: by splitting powers like that, foreign actors need to consider internal US politics to a large degree. Stability would be greater if a simple majority in the house had the ability to appoint a new president at a whim - not because whims are stable, but because that shapes the back and forth of political powerplay, and thus renders the house responsible for the presidencies errors, which is how it works in other countries. Countries don't tend to reappoint prime ministers all that often, especially if a coalition is involved, because not only is that likely to be punished by voters, it's quite hard to form coalitions if you break support like that. That's not to say that a coalition of microscopic parties would be a great idea for the US by the way; just that small steps in that direction would increase stability.
Wouldn't forcing senate reelection in case of failed impeachment allow the party that controls the house to force that reelection?
Let's say Party A controls the house while Party B controls the Senate and the Presidency. If Party A wants, they would be able to send impeachment to trial without real reason and then by the fact that Senate didn't convict you would get a reelection of Senate.
Way I see it, this would allow manipulation of the election process. In my mind, it would be much more logical that the House has to be reelected as they pushed forward an impeachment that was not convicted upon. This would make it so that House would not bring forward frivolous impeachments as they would have their own positions on the line.
About unseating a president, I understand your point. I agree to a degree. A sitting president can be limited in power by the fact he doesn't control the legislative branch. This may seem less optimal but it has a couple of advantages. Executive branch can keep on working within the confines allowed to them, without having to juggle for political support in the parliament. This allows the President to keep working with little regard to losing his position. This is especially good for military or foreign affairs as they can work on this with little to no political support.