Zmejka, the answer is that all of them might be acceptable for different compounds.

Internet related vocab is a classic example of how language and its orthographic representation develops to reflect the frequency of use and how established certain terms become in our daily lexicon.

There is a general trend for word formation where two words become one over time via an interim period of hyphenation!

Thus horseman probably started life as horse man and then spent a period as horse-man - do you see what I mean?

Sometimes you will find that 2 or even all three versions are acceptable usage at the same time, and that preference will vary depending on house style, geography etc.

But to be specific I would say that for the examples that you provide, the most common spellings (in the UK at least) are:
website (through frequent usage it has earned the status of a word in its own right).

web design (two words)

As for others:
webmaster (one word, a neoligism with strong precedents in words like postmaster).

Web page (two words - sometimes it is also I think to do with how well two words mesh together phonetically - b and p are awkward to pronounce in one word)

So to summarise, there is no single rule, but there are various regular factors which might influence which you choose.

Hope this helps