Getting Better - Comparative Adjectives

A single word is clearest, though 'more cloudy' is possible and certainly not wrong.
Any other two-syllable adjective, where the second syllable was a -Y, can form its comparative version this way: rainy/rainier, silly/sillier, happy/happier (and indeed its opposite compound, unhappy/unhappier ~ though many people, if they need to, will more likely say 'more unhappy').
And don't forget: even though it rains a lot in Britain, in general our summers are DRIER (= more dry) than the other seasons.

'More' is fine on its own here.
English has a saying, 'The more the merrier' ~ meaning (usually) that the more people there are, the more fun they will have. If someone offers you a lift in a car, and there are already quite a lot of people inside, they may use this phrase, but you might prefer not to ride along if you feel it could be dangerous ('too much of a good thing')!
Sometimes people say this about putting alcohol into a 'punch'-type mixed drink. You may prefer to be careful then, too!

'Good ... better ... best'!
In English we do put 'more' in front of an adjective (usually if it is longer than one syllable), but 'better' is an exception, and we are sure you will hear and see it often.

Most adjectives that are more than one syllable long (and don't end in -Y) will form their comparative version by staying the same but putting 'more' on the front.
French, for example ~ and its sister languages ~ can only ever do comparisons this way with most of its adjectives ('X est plus cher qu'Y'; Spanish 'mas grande' etc.); whereas German can just say 'X ist teuerer'.

Answer 1 is the best, though Answer 2 is also possible.
'Tasteful' (Answer 4) doesn't quite mean the same thing: it refers to having good taste, in the sense of not preferring such things as very loud music, or violent / clashing colour-schemes. (Most people, for instance, would probably prefer a 'tasteful' funeral or other ceremony when they die: an event with respect and dignity, rather than a great ugly drunken party.)
'Deliciouser' doesn't work, though English people would understand what you mean, probably smile, and remind you gently that we don't make -ER -type comparisons with such long adjectives.

It's a bit sad that we have to have such a word at all, but sometimes we need it. There seems to be no link between 'bad' and 'worse' in the actual shape of the word; they don't even have one single letter in common.
Obviously, if you have a problem while you are staying/working in Britain (for example, a leaking tap in your bathroom), you will need a word to explain that things are no longer as good as they were!

'Nicer' is possible, but a dull and over-used word; 'pleasanter' may be possible, but 'more pleasant' would be a better form.

'Further' is the more usual form; sometimes you may hear 'farther' (usually in a more formal context, or from old-fashioned people ~ and not to be confused with 'father', which sounds rather similar!).
Actually, you would hear a difference between 'father' and 'farther' if you were listening to an English speaker from the West Country, or many parts of Ireland, Scotland and elsewhere, or indeed from most of the USA. Reasonably enough, they do pronounce the R distinctly, so you would hear: 'My pooR old gRandfatheR is too tiRed to walk any faRtheR this afteRnoon'!
The problem is that speakers of 'standard southern English' tend not to pronounce the R clearly in such situations (otheRwise it would be 'standaRd southeRn English'!).

'Elder' is the form that English speakers use widely when referring to people, though 'older' is still correct.
We do not use it to describe anything other than people (or just possibly, animals: 'the elder sister of our dog went to live with some friends in Bristol'). We would say that Oxford University is older (not 'elder') than the one at Cambridge. In fact, we wouldn't use 'elder' in a normal comparison of human beings either (we wouldn't say 'Paul is elder than Fred'), but we might say Paul is 'the elder of the twins by 20 minutes'.

'Quick' is a normal single-syllable adjective, so its comparative form is regular. It would somehow be odd if the word for this idea were long or diffficult! (Though, in common with some other languages including French, our word for 'long' is actually shorter than our word for 'short' ~ 4 letters instead of 5!)