"There are four pictures."
写真が四枚あります。and 四枚写真があります。 are both correct. In Japanese, number + counter either goes in front of the verb or in front of the object. However, I noticed Duolingo often objects to the latter. The course is in development still, so until this is fixed, best stick to Object - Number + Counter - Verb.
つ, as opposed to 枚, would literally mean "things" where in your case it might work, but not as effectively. Most of the time in English if you used a camera to take pictures and put them on your computer, you would say "There are 4 pictures" instead of "There are 4 things." You still understand it but the second one seems a little less specific.
四つ - Four (object counter) pictures 四枚 - Four (thin object counter) pictures
The difference is in the counters
つ is an object counter you can use when you don't know any other counters for that object (I might be wrong).
枚 is a thin object counter - the picture here is meant as a photo printed on a piece of paper - paper is thin, so you'd use the 枚 counter.
When you translate the correct sentence, 絵が四枚あります literally means 「絵」Pictures 「が」as for the (subject marker) 「四枚」four sheets 「あります」there are. You put it altogether and it literally translates to "As for the pictures, there are 4 sheets."
On the other hand「を」means whatever this particle marks, the verb is doing that to the noun. The verb in this sentence is 「あります」which means to exist. We aren't existing 4 pictures, we are having four sheets, which are pictures.
ある（あります） means to be or to exist - "There is/are". If you wanted です, the English equivalent would be "It is/Those are" as です means to be. And "for people doing actions" - I think you mixed that up with something completely different, although I'm not sure what exactly. If you want to say that "There is/are (a living person/animal/etc.), you'd use いる（います） instead of ある（あります）.
あります - used only for inanimate objects います - used only for animate objects
Not very, though if there is a specific counter used for something it's a bit strange not to use it. Like saying "Four things of paper" instead of "Four sheets of paper".
(You would use が with the existence verb あります though since it isn't a transitive verb/does not take a direct object)