It's a Lovely Day - Weather Words and Terms

If the forecast warns of 'showers', it means they expect the rain to keep starting and stopping as the clouds pass over any one place. ('Showers' ~ plural ~ because there may be several 'bursts' or episodes, and you may not be able to tell how long each one will go on or when the next one may start.)
This is not quite the same thing as the 'shower' in our bathroom, although you can surely see the connection. Many other languages use quite separate words for these two things (e.g. French: 'averse' / 'douche').
A typical feature of British weather, especially in spring and autumn ... but not only then, necessarily!

We can also 'melt' other things which are solid at normal temperatures. If you are very lucky with the British weather, you may go on a picnic and find that your butter or chocolate has melted, or 'gone runny'.
If there has been snow and/or ice for a long time and it melts at last, we sometimes say it is 'thawing' or 'has thawed'. We also use this image occasionally to talk about a relationship where people stopped speaking to each other, and then, perhaps some long while later, began again.

Names of months in English always start with a capital letter, and we say 'it is windy' (like 'it is hot / cold / foggy' etc.).
Unlike many languages, English uses 'it is + (adjective)' a lot more often than 'there is + (noun)' (Spanish: iHay viento!).

Drizzle is the sort of rain where the drops are quite small (or 'fine') and they don't seem to be falling hard or quickly. But your clothes will still get surprisingly wet if you go out in it without any protection!
Of course drizzle is relatively 'soft (Answer 3), but it is definitely 'wet' like any other rain (Answer 4) ... how stupid!

This phrase usually means that the sky will be cloudy ('overcast') for more of the total time than when it will be clear and bright ~ perhaps 20 minutes of sun here and there, and not again for another couple of hours. Very typical British weather, in fact!

Adjectives like 'happy' (and 'sleepy' and 'funny', and all those other ones that end in -Y, like the Seven Dwarfs with Snow White!) form their comparative by dropping the Y first and then replacing it with -IER. If there is a double consonant towards the back (here, -PP-), it stays doubled.
When we make the -ING form of a verb that ends in a silent -E, we take the -E off first ('shin-' + '-ing'). We do NOT double the consonant, because if we did that, the first 'I' would become short : it would sound like 'shinn - ning', which means something quite different.

'Hail-stones' = 'stones of hail'; the 'storm' (Answer 1) is the whole bad-weather event that brings you the hail.

( ' it already is ... raining' )
In English we quite often shorten-off a sentence once it is clear what the repeated verb would be.
('She asked me to open the door, but I already had [opened it for her].')
If it's raining already, that would explain why you can't see the faraway tree ... because the sky in between is half-full of drops of water!

'Rain' is singular (even if there's lots of it), and tomorrow is in the future, so this is the only possible answer. You will hear such things a lot on the weather forecast on the radio or television.

Nothing can move more quickly than light, so the lightning will come to you first (unless you are directly underneath the storm point, where you would experience them more or less at the same moment: exciting!).
In English, you can hear something without (actively) listening, and you can see without (deliberately) watching. If a storm is going on, it's probably so strong that you can't stop yourself knowing about it. It would be hard not to hear a clap of thunder, or not to see a flash of lightning. So, apart from some other spelling and style mistakes, Answer 4 was not at all good.