In most multi-party systems, parties are always forced to compromise and cooperate to get anything done, since it's almost impossible to get a single party majority. Dutch governments are almost always formed by 2 to 4 parties (usually a centrist party plus either left- or right-leaning parties) which largely prevents swings to extremes.

No, it means the natural distribution of views gets turned into a bi-modal system of allegiances. Each party defines itself by its opposite. It then becomes much harder to cross the gap because doing so involves changing most of your political views to their polar opposites and losing most of your friends.

Quantifying "social division" is tricky, but I feel that multi-party systems offer much more scope for saying "we agree on A and B but disagree on C". While small differences can also be very divisive because humans are like that (big problem for the Left) it means you get less negative voting.

"Against X" and "For X" are not symmetrical positions, because there are usually lots of different alternatives to X. Only in a 2-party system do they produce the same answer.

The UK is mostly, but not totally, a 2-party system, and it's really noticeable how the regional parliaments do politics in a completely different way, with different party groupings. This is at least partly to do with the voting systems (AMS for Wales and Scotland, STV for the defunct NI assembly).

Two parties means one or both can go to extremes pretty easily. If both do, people just pick one and the parties use votes to justify their extreme views. If only one is, it usually leaves the second party simply fighting against insanity (I think this is part of what we are seeing now: Bernie might be the start of a more extreme opposite to the other party). The parties also don't really have to have steady standards or party stance on things, at least not in the states. It more depends on the individual.

Multiple parties mean that more views can be heard. You can have a party with extreme views, but people can choose something different. Hopefully, there is a bigger range. The other thing multiple parties seem to encourage is working together and compromise. If you've 5 or 6 parties, it is harder for one party to hold up the government.

It isn't to say that a two-party system can't be what you describe, but it certainly isn't what is happening in the US right now.

In a five party system, it is quite common for voters to not feel too much allegiance to any party in particular.

That depends on what you determine is important to you. In the Netherlands there is a 'Party for the Animals', which doesn't compromise on anything and is quite literally a protest one-issue party. Obviously they don't get as much done as other more diverse political parties, but people still vote for them because the party at least shows everyone else that there are citizens who care enough about animals to have a protest vote and have the chance to at least sit in congress.

Exactly, here in the Netherlands you still need a majority, this is often formed from 3 or more parties that have to negotiate a lot before coming up with a plan.

Having only two options forces the two to be different, more options provides you with a better spectrum of political ideology.

It also depends on the way the two party system is organized. One point that keeps the American system as polarized as it is is closed primary elections, which guarantee that only the most fervent and 'ideologically pure' candidates are selected. Since only registered party voters can cast ballets, the type of candidates that resonate are elected and continue on to the national election.

In the last elections for example I heard multiple times from friends and family (that identify more with the democrats) that they would have considered voting for someone like Kasich (2016) or Jon Huntsman Jr (2012), but are robbed of that choice because they are blocked from voting in the republican primary.

I don't think so, because you might agree more easily with "other people" on several issues and therefore switch more easily to another party.

The two party system sucks in all moderates, and forces them to support the most radical positions.