1. What is a paragraph?
A paragraph is a collection of sentences which relate to each other
because they deal with the same topic. Writers divide their work into
paragraphs because paragraphs separate out different parts of an argument
and make the argument easier to follow. Think of paragraphs as a kind
of route map through an essay.
2. How many sentences make a paragraph?
There isn't a fixed answer to this one. Different kinds of writing use different length paragraphs. At one extreme, a tabloid newspaper tends to use very short paragraphs, and it often makes each sentence a separate paragraph of its own. In academic writing, of course, you shouldn't do this - unless you want your essay to read like something from the Sun or the Daily Star. At the other extreme, old-fashioned or very formal writing tends to use much longer paragraphs. I've just looked in Thomas Malthus's Essay on Population, a book published in 1798, and I've found a paragraph which is 34 lines long and has 15 sentences in it. This is too long for modern readers - and 'modern readers' includes the tutor who will mark your essay.
As a rough guide, you should aim to build paragraphs which are at least four sentences long, and not more than eight sentences long. As I said, there isn't a fixed rule, and you can vary the length of your paragraphs. In fact you should vary them, because if all your paragraphs are the same length, the essay will become repetitive to read. But try and stay within the 4-8 range for most of your paragraphs.
3. How many paragraphs per page?
The answer to this one depends on how long your sentences are, and there isn't a fixed rule about it. This is just a rough guide, but try to break up each page of print into around three paragraphs. A page which is all one paragraph would be about 500 words long (I'm assuming you're doing single-space print, like this) and that would be hard work to get through in one go. A tutor reading a page like that would be likely to think that you hadn't thought enough about how to structure your argument. At the other extreme, if you break up each page into ten or more paragraphs, it looks as if you aren't developing each point in enough detail.