I have not heard the expression 'play the cubic'. When I used to play it, I would say either I am playing the Rubik's cube OR I am playing with the Rubik's cube. The first one has a sense of playing a game, and the second the sense of playing with something for fun or to work out how it works.

They are all fairly similar :-) Some differences are grammatical, for example, Fairly can be used with adjectives and adverbs ('it is fairly good', or 'I did fairly well in the test') However, 'quite' can be used with Adj, adv, and also verbs and nouns 'I quite like this' (verb), 'you are quite a chef' (noun) I would also say 'quite' is stronger than 'fairly'. if I say 'this is quite a good book' - it is a stronger recommendation than if I say 'this is a fairly good book'. I think 'rather' and 'pretty' are stronger than 'fairly' and 'quite', and may suggest something more than what you expected. For example, 'this book was rather/pretty good', suggests you are pleasantly surprised, it's better than you originally thought. Hope this helps!

'Your finest hour' refers to the moment in your life when you make your greatest acheivement or have your most success. In this sentence (taken from Radio Ga Ga if I am not mistaken), it means your moment of most success has not yet happened, and will happen in the future. Given the recent success of podcasts, maybe Queen were absolutely right!

Special dinner with my husband

We had the special dinner which was the same as the wedding menu for our guests. We held our wedding ceremony in the autumn so we could feel a sense of the seasons from the different cuisines. Every dish was amazing and beautiful! They all made us feel satisfied.
We became fans of the bland of Ukai. I want to go to another concept restaurant with my husband.

I am not completely sure of the tenses here because it is not clear if the dinner you describe is at your wedding or some time after. I have  assumed it is some time after.

I think the first one, simple future to show something that exists (you being free) for a period of time (until 6pm) into the future.

I LOVE LONDON

I visited London on the 31st of December, to celebrate New Year's Eve in a city that I have always loved since I was a child.
The first minutes I saw the big city, I was shocked. Everything was different from my hometown. The people were nearly all blonde, there were beggars here and there, the houses were all so perfectly 1.shaped, and a little bit old and classic.
The streets were so elegant with a touch of extravagance.
The sky was always grey, but sometimes the weather was sunny and at the same time cold; fortunately it never rained the whole time I was there.
The food was terrible and too expensive, much better the Italian food!
I loved how London could be so different from one street to another, it was like a melting pot full of different people, tourists, artists, museums streets, and parks... 

I was amazed by the fact that the underground was so well organized and it was always on time. It can take you wherever you want to; a lot different from my country, where trains are always late and the underground is always jam packed.
The spot that I liked the most, was the view of the Thames from Waterloo bridge.
When I was there I missed my home, but when I returned home I wanted to go back there.
I think that London has the power of letting a stranger fell in love with the city.
London is so fascinating, so charming, so much everything...
I wish I could go back there 2. sooner or later. 

Great writing - lots of feeling about a lovely city!

1. Not sure what you mean by 'sharpened' - I put 'shaped' but maybe you meant something else!

2. Or you can say - 'sooner rather than later'

1. Sounds a little strange - we may 'dig out' something old or that has been hidden away, that I now want to use, but doesn't really work with 'privacy'. 2. Sounds natural - although I would probably use 'private matters' or 'private business'. The use of 'Privacy' as a noun sounds a slightly formal and clashes a little with the informal 'stick your nose into...' 3. Same as 2. Other ways - Why are you always trying to snoop into other people's personal business? Why are you always trying to poke your nose into other people's personal business?