Translation:I think that test is probably easy.
Casual forms for nouns,
です = だ ，
ではありません = じゃない ，
でした = だった ，
ではありませんでした = じゃなかった 。
As others have pointed out - not actually the casual form of です. Better thought of as a declarative marker at the end of the sentence that makes it assertive (thus reducing politeness). In some relative clauses, you must use だ to help disambiguate the sentence. Sometimes だ can be used in place of です, but it really is better to think of them as two separate beasts.
This link might be handy: http://www.guidetojapanese.org/learn/category/grammar-guide/page/5/ (see section "「です」 is NOT the same as 「だ」")
Not exactly the same. For instance you could not use desu in that sentence, but you would need to use da, even if using the polite form. It's worth looking into more. For instance, even in casual speech, saying sou da is more forceful than sou desu. In this case, unless you wanted to really emphasise your point, you'd just say sou. Just remember that even in casual speech, you can't always just switch desu out for da.
Why? I don’t think, that the original sentence possesses that kind of uncertainty.
"might be easy" ≠ "is easy"
I'd agree, "I think it might be..." is much more natural than "I think it's probably..."
As I understand it, you don't put polite forms in the middle of sentences, is that correct?
Exact words: "The final verb indicates the politeness and tense of the whole sentence."
If that's so, you're only ever putting masu and desu at the end.
As a general rule that is correct.
There are some exceptions, namely when the first half of a sentence is a subordinate clause, connected to the main sentence by conjunctions such as が (but), ので or から (because). In a way, you're still only putting ～ます or です at the end of a sentence, it just might not be the final end of the sentence.
E.g. 友達（ともだち）と一緒（いっしょ）に公園（こうえん）に行くつもりですが、彼は風邪（かぜ）をひいていますので、今日は行けません。 -- I want to go to the park with my friend, but because he has caught a cold, we can't go today.
It's a quoting particle. Try to think of the sentence like this: "That test is probably easy," I think.
"That test is probably easy I think" Marked incorrect. Getting mighty tired of this ❤❤❤❤, duolingo
To me, 'I think that test will probably be easy' is a better sentence. Correct me if I'm wrong.
The way the audio pronounces the two supposedly seperate words たぶん and 簡単 makes them feel like one phrase instead.