Very true - and similarly for all sorts of other common contractions like να'σαι, σ'αγαπάω, από 'δώ and so on. The current tree doesn't cover contractions though and I think they're probably best learnt at a slightly more advanced stage when you know what letters are being dropped and why.
That's true; the above sentece would be κλίκαρε 'δώ and not κλίκαρ'δώ, however. The rule is that one vowel "eats" the other, not that they both disappear.
Yes, but when two vowels are present (as in "Κλίκαρε εδώ) it's not correct to omit them both.
πιο συχνά λέγεται ΄΄πάτα΄΄ εδώ και όχι κλίκαρε.