Your English is quite good.  I would assume that you do not seek to remain at the same level and seek detailed corrections.  Therefore, I will pick this apart for you.

The Most Uplifting and the Biggest Downer (Capitalize important words in titles)

-  ( referring to yourself here is unnecessary because you use "me" later in the sentence.  While it isn't incorrect, it sounds awkward to have it there twice.)

The film is so action-packed, that it made me sit glued to the screen. However, it has a lot of sickening scenes and sometimes it  is really appalling.

- (You needn't use "it" a second time for the same subject in the same sentence.  Example: It has an apple and is eating. Two things that are happening without a comparison.  If you look at my earlier correction, there is a comma which would require the use of the subject again.)

Unfortunately, the film turns out to be very depressing and leaves little room for hope and this would definitely put me off watching it again.

- (This is either a list that was not separated with a comma or a run-on sentence with so many "and"s.  You have two options here: make the description into a comma separated list or split the sentence into two.

   1. Unfortunately, the film turns out to be very depressing, and leaves little room for hope, and this would definitely put me off watching it again.

-You would replace the first "and" with a comma.  You would, also, remove the word "this" because it has now become a list about the film and "this" would no longer refer to the reasons you listed, but the film now putting you off.

    2. Unfortunately, the film turns out to be very depressing and leaves little room for hope. This would definitely put me off watching it again.

- In this option, you replace the second "and" with a period.  "This" is now still referring to the reasons you listed as being the reasons why you would be put off from watching the film and needs a new sentence because it is a different object.

“The Eternal Sunshine of the Spotless Mind”, has a way (not incorrect, but a very informal descriptor.  "an incredibly" "drastically" "presents quite a different") different atmosphere.

It explores the nature of memory and romantic love and would appeal to everyone who loves romantic stories.

- (Nothing is wrong here, but I will explain because I said the earlier sentence had too many "and"s.  This sentence uses two ideas "It explores and would appeal".  The first idea uses a two item list (ex. I have apples and bananas.) It is entirely correct to use two "and"s in one sentence when one is for a joining items in a list and the other is for joining two ideas.

This story depicts two former lovers: (colon) Joel (Jim Carrey) and Clemetine (Kate Winslet) who, after having spent two years together, (another comma) try to come (In English we "work" or "go" through breakups) through a sickening break-up. When Joel receives a note saying that Clementine has had him erased from her memory, Joel decides to do the same. As a medical crew enters Joel’s brain to hunt down his memories, he realizes, too late, (commas) that he wants to keep them.