This notebook entry is fully understandable and has some good use of casual expressions that fit the desired format of an informal letter.  I would make the minor changes shown below:

4 weeks to get ready. (informal letter)

Hi Jo.
4 weeks aren't many at all is not much time, but as you don't start from 0, you will be able to succeed if you are motivated. Let's see what you can do.

Firstly First, get in touch with my friend Davide. He studied sports sciences at the university and now are is working as a personal trainer in a gym as well as a coach in on a basketball team. He may definitely can suggest for you a great training plan to spoil these few days at the best to use your time to best advantage. If you don't know him well, don't worry.  Mention my name. If you mention my name, he will be glad to help you.

As far as I'm concerned, if I were you, I would change my diet immediately. You must cut down junk food and sugary drinks and go for food foods that helps help you to get in shape. You can't afford to allow yourself even a beer. You will probably work out every day so don't forget to rest. You can't do your best without recharging your body.

I'm sure you will make it. Let me know how it goes.

Go to running I'm going to now!  (If you are really going running).  Got to run now! (If you are just using this as a closing to your letter.)

Paolo.

Notes on a few of my changes:  

"4 weeks is not much time" sounds as if it is incorrect pairing of a plural subject "weeks" with a singular verb "is".  Actually, "4 weeks" is treated as a label or name for the single unit of time allotted by Jo.

"may definitely" juxtaposes a conditional state with an absolute state - it is an oxymoron. 

"If you don't know him well . . . " is a logical flaw, since you are making the introduction yourself.

These are rather detailed comments, but I judge from the generally high ability level you have shown that this is your desired level of scrutiny.

All four statements sound perfectly normal to a native English speaker from the U.S. The options that add "thing" into the thought are more common, but I consider them less articulate, as they add unnecessary, inexpressive verbiage to the expression.

Translation help beautiful couple.

You guys are a beautiful couple! We miss you both. See you soon!

Voi due siete una coppia bella.  Ci mancate a tutti e due.  A presto!

Sunshine's comment is what would most likely be encountered, exactly because the grammatical construction seems awkward. Native speakers would likely do as she has suggested and work around needing to use the possessive form. Vanessa also suggested the same idea - to not use the form at all.

A student's "major" at a university is a group of courses which make up a defined program of study designed to demonstrate both scope and depth in a field of study.  It usually includes 6-7 courses that are expressly required by the academic department, and then also requires 3-6 courses that are electives, i.e., courses chosen from a selection of 10 or more possible courses available to supplement the core requirements.

Here is my answer, which I incorrectly added above as a comment: As you can tell, native English speakers have trouble with this. Common, casual American speech would avoid the problem altogether, and say: "Tom is friends with Jim and me." The grammatically correct construction is: "Tom is Jim's and my friend." The handy practice aid is to state the question without the intervening name, and just use the pronoun: "Tom is . . . my friend". Or "Tom is friends with . . . me." Then simply insert the other name in the sentence first.