You're going to run into issues with the grammar and structure of Japanese being much further from English than Spanish is, so the same methods that worked for Spanish may not work as well for Japanese. In particular, if you haven't been taught how to use a grammar structure first, trying to get you to translate direct from English is unlikely to work out well. Look for material that teaches you the grammar first, then gets you to form lots of sentences with it for practice.
Erin's Challenge is I think the best of the online resources. The people who run it also run the JLPT, and there's lots of examples (video skits, manga, etc) alongside the explanations.
Okay, a quick run through your questions:
達 & たち are the same - 達 is just the kanji form. Many times the choice is simply that - a preference. People try to get a nice balance between kanji and kana - a lot of kana with little kanji is more casual and can look childish (since children don't know as many kanji), whereas over-use of kanji can look stiff and old-fashioned. But there is a lot of room for personal preference between the two extremes.
たち is not really a "pluraliser" in the same way as we pluralise in English. It's more of a "grouper". For this reason, you don't normally append it to objects unless you're being poetic (so オレンジたち would be weird, 星たち you might see in fiction or song lyrics or something). Similarly, you can use it attached to names to mean "so-and-so's group", e.g: 田中たち doesn't mean "multiple Tanakas", it means "Tanaka and others". e.g. if Tanaka and some others are going to come to the party together, you might use 田中たち to refer to that particular group of people.
Use たち then not when you mean plural, but when you're identifying a particular grouping of people. So they shouldn't have marked 私達は男たちですか as correct.
In Japanese, context is used to differentiate plural and non-plural.
e.g. 大学生 would mean university student or students, depending on the context.
彼は大学生です。 = student, since 彼 refers to a single person.
私たちは大学生です。 = students, since 私たち refers to multiple people.
大学生は金がないです。 = students, since we're making a general statement about them.
数人の大学生 = students, since we've appended "a few".
一人の大学生 = student, since we've specified there's only one
On your others:
"masu" form is politer. Which form you use depends on who you're speaking to. A very general rule would be plain form for friends, desu/masu (also called 丁寧語/teineigo) for people you don't know well or those senior to you. There are more levels of polite language (keigo), but you really don't need to worry about those for now.
~ている or ~ています, is used for either an ongoing action (I am eating), or a state (I am married). If someone currently has a newspaper, therefore, you should use this form. The plain verb form is used for future (I will get married) or habitual (I eat breakfast at 7am every day). 彼ら is also weird here unless a group of people have one newspaper? Is that not just the gender neutral "they" in English?
いや、違う、そうじゃない!
いや = "no"
違う = literally "different", but used as an assertion to mean "the thing you just said is not correct".
そうじゃない = "it is not like that".
So not "no" as in response "Do you want some coffee?" but "no" as in response to "Are you secretly in love with Tanaka?"