What those two bodies of government represent, and what is their respective responsibilities, in simple terms?
Both the House and the Senate need to vote affirmatively on legislation for it to pass. Only the House has the power to introduce bills involving revenue, and it also has the power to impeach officials with a two-thirds vote. Only the Senate can ratify treaties and confirm the appointments of federal officials that the President appoints. (Like Supreme Court Justices, which is a relevant discussion right now) The number of congressmen and elections are different between them as well: The House has 435 members elected from districts that are divided between the states by population, whereas the Senate has 100 members, 2 for every state. Senators are usually more moderate than representatives, probably because they represent more people.
Why do only the two parties matter? Solely because they're the biggest (I hear a lot about them and barely anything, even in history, about any other)?
It's hard for any third parties to spring up because of the first-past-the-post system that is used in (as far as I know) in every election here. And you're not wrong with the "biggest" thing: the two major parties are both much bigger and more influential than any smaller parties.
What does the partition mean for the country's political atmosphere? What's those positions', party-wise, influence on the country so far - that is, for example, how is a Democrat president different from a Republican president? Does it even matter what party they are? Does it matter if the President is an unaffiliated person, parties-wise, and why if it does?
Going to answer these questions out of order, so bear with me. There hasn't been a president elected as an independent since George Washington, so a candidate managing to do it would be essentially a first. It matters a great deal which party a president belongs to because the parties are so ideologically split: a Republican president would work to support right-wing legislation, would try to appoint right-leaning officials, and so on. A Democratic president would do the reverse. In addition, unless it passes both the House and the Senate with a very large amount of support, legislation can be vetoed by a president. This also leads into the current state of the split between Congress and the President: Congress can pass all the conservative legislation it wants, but the President can veto it whenever he wants, because the Republicans still don't have a veto-proof majority. They've been pretty much at an impasse since the last election in 2014, when the Republicans took over the Senate.
Why do they hold the first primaries? Why is the order of the primaries as it is currently? Clearly, it's not alphabetical, and I don't have a map of the US to confirm that it's geographical.
A lot of it is tradition, and a lot of it is state law. Iowa and New Hampshire, confusingly, both have it in law to be the first primaries. Since they use two different systems in their primaries, they can both technically be "the first". But if any other state tried to make their primaries before them, they would simply move their own primaries up. The order of the rest of the states changes year by year, but typically a lot of states cluster their votes on "Super Tuesday", which is a Tuesday in either February or March.
What about the people? What does the general population think of her? I don't know of her political agenda or promises, but so far, she gives off a vibe of a person I'd trust to run my country had I been a US citizen.
She started out with a lot of support among Democratic primary voters, but she's been losing a lot of support to Bernie Sanders. Her strength among the elite has remained secure, which is why I only mentioned that. She's pretty popular with Democrats, but she's unpopular with independents and hugely unpopular with Republicans for a lot of different reasons.
Why do Republicans have so many presidential candidates? Is it because the last president was Democrat?
After John Mccain lost in 2008 and Mitt Romney lost in 2012, there was a huge gap in Republican presidential candidates, and everybody who thought they had a chance jumped at the opportunity. Most of them failed pretty hard already.
Finally, is it "president" or "President" in English?
I use either one depending on when it feels right. I honestly don't know the proper capitalization.
Hope this helps.