For context I understand the American political system quite well, and have a somewhat weaker understanding of how the UK government works. With this disclosure...
My understanding is that the basic difference is that in the United States the executive head (President) is elected through the convoluted process of the electoral college, but in places like the UK the executive is selected from among the legislature. If we were to make an analogy it is like there is no President, but instead the Speaker of the House, roughly analogous to the prime minister, is the president.
So far so good. Next is how are the legislators chosen. In the United States the country is divided up into physical geographic regions of "roughly" equivalent size and representatives are elected, one from each district, and in the UK this also seems to be true, in that each MP represents one physical geographic region.
First question - is the above a reasonably accurate summary of how the UK government works?
Second question - the UK uses a first past the post system to select someone for a single seat, which as far as I can tell is basically exactly the same as to how Americans elect our house of representatives. Winner take all systems like this tend to produce 2 parties, and while the UK certainly has 2 major parties and a bunch of smaller parties in the US third party candidates basically never win anything, whereas in the UK there are always a bunch of "third party" candidates that actually get seats in the house of commons. Why are "third party" candidates so much more successful in the UK as compared to the United States? In the House of Commons it looks like there are something like 12 different parties, and 3 different parties hold at least 10% of the available seats, whereas in the United States I don't think a third party has held more than 1 or 2 seats in the House of Representatives in decades.
Last question is why holding new elections would be likely to change anything. If parliament doesn't elect a prime minister one of the things that can happen is new elections, but what I don't understand is why that is likely to help anything. I feel like in the United States if, during the time when the House couldn't elect a speaker we had instead re-done the house elections what we would have ended up with would be the sending the same group of people back to fail to elect a speaker. When parliament is dissolved for failing to form a government why is there any reason to believe that the new parliament is going to be any different from the old parliament?
Is the difference the fact that in the UK you can't vote for a "split ticket"? In the United States you can vote one party for executive and another party for legislative representative, but in the UK since the prime minister is not elected directly but is rather elected by parliament the only way you can get rid of a prime minister whose policies you hate is to get rid of your representative who put that prime minster in power?