I need a few things for the happiness = Good idea here. But, "Only a few things are needed for happiness"/"Just a few things are needed to make me happy". These would be the more native way to speak depending on the meaning you want to get across. "Only a few things are needed for happiness" is for a general population/ general blanket statement that can apply to anyone; therefore, the article [the] is replaced with the [zero] article. "Just a few things are needed to make me happy" is specific to you which is accomplished by the indirect object pronoun [me]. In my 2 examples [only] and [just] are interchangable. Also, "I need a few things to achieve happiness" works which uses most of your original words, but it introduces a different meaning. It means that you are kind of hard-to-please because you got to have more than 1 thing to be happy. 

I felt I could contribute most to this one issue since Caterina and Neil have already done a great job explaining other points.

Best. Message me anytime.

I will assume that "Nataly" is a woman.  Therefore, you need to use the feminine possessive pronoun "her":

This weekend I met my friend Nataly and her sweet little daughter.

I recommend saying "sweet little daughter" rather than "little sweet daughter".  It sounds better to me, but maybe it's just a question of taste.

I need a few things for the happiness.

Incorrect expression: try "to make me happy" or "for happiness to find me."

This weekend I met my friend Nataly with his little sweet daughter.

Adjective order: personality + size + noun

We went for a walk among flowering apple-trees.

The sun shined, light breeze blowed round our faces.

Incorrect past tense of shine.

Change to two sentences or use a conjunction to join the ideas.

Then we sat talking on the gaudy plaid under a tree. Nicely structured sentence! Interesting description. Can a tree be gaudy?

It was amazing weekend and we was absolutely happy!

Missing article.

Incorrect past tense of to be verb..

I hope these tips help you Anna. Caterina :)