Can you tell me if the following sentences or phrases are possible? Thank you very much in advance.

1) I'm wearing a pair of blackt trainers withe the logo Nike on them,
2) Did you have to work this weekend?
3) I usually wear patterned, checked or striped T-shirts over a pair of blue, skinny jeans.
4) I wear T-shirts of every (all?) kind.
5) The "be going to form" is used either to express the speaker's intentions or to make predictions based on evidence.
6) On the other hand, the present continuous is used only for planned future actions and is generally followed by a time expression.
7)You mustn't speak up in the library.
8) No definite article before public institutions such as school or college.
9) You must (?) be careful! You are going to fall.
10) Do you say "tuxedos" to indicate the plural?
11) Do you say "a long-sleeved T-shirt or a "long-sleeved shirt"?
12) a long-sleeved T-shirt or a T-shirt with long sleeves
13) a T-shirt without sleeves or a sleeveless T-shirt.

Again, delete everything in parentheses.

I don't understand #5.

#6 is grammatically correct, but the content isn't -- present continuous is for indicating present actions; future continuous is for future actions.
http://grammar.ccc.commnet.edu/grammar/progressive.htm

#8 is not a complete sentence.

#10 http://www.answers.com/topic/tuxedo
(yes)

#11-12 -- either will do, depending on the shirt. The term "long-sleeved shirt" usually refers to a shirt with a collar, while any T-shirts do not have collars.

#13 -- the 2nd one is better.

1) I'm wearing a pair of black trainers with the Nike logo on them.

2) Did you have to work this weekend?
3) I usually wear patterned or striped T-shirts over a pair of blue, skinny jeans.
4) I wear many different kinds of t-shirts.
5) The "be going to form" is used either to express the speaker's intentions or to make predictions based on evidence.
6) On the other hand, the future continuous is used only for planned future actions and is generally followed by a time expression.
7)You mustn't speak up in the library.
8) You don't need to use definite articles before public institutions such as school or college.
9) You should be careful! You are going to fall.
10) Do you say "tuxedos" to indicate the plural? (yes)
11) Do you say "a long-sleeved T-shirt or a "long-sleeved shirt"? either
12) a long-sleeved T-shirt or a T-shirt with long sleeves either
13) a T-shirt without sleeves or a sleeveless T-shirt. 2nd one

Sure! I'll go through each sentence or phrase and explain if it is possible or not.

1) I'm wearing a pair of black trainers with the logo Nike on them.
- This sentence is possible. It describes the speaker wearing specific black trainers with the Nike logo.

2) Did you have to work this weekend?
- This sentence is possible. It asks if the person being spoken to had to work during the weekend.

3) I usually wear patterned, checked, or striped T-shirts over a pair of blue, skinny jeans.
- This sentence is possible. It describes the speaker's typical choice of wearing patterned, checked, or striped T-shirts with blue, skinny jeans.

4) I wear T-shirts of every (all?) kind.
- Both "every" and "all" can be used in this sentence, so it is possible. It means that the speaker wears T-shirts of various types.

5) The "be going to" form is used either to express the speaker's intentions or to make predictions based on evidence.
- This sentence is possible. It explains the usage of the "be going to" form for expressing intentions or making predictions.

6) On the other hand, the present continuous is used only for planned future actions and is generally followed by a time expression.
- This sentence is possible. It explains that the present continuous tense is used for planned future actions and is usually followed by a time expression.

7) You mustn't speak up in the library.
- This sentence is possible. It expresses that speaking loudly is not allowed in the library.

8) No definite article before public institutions such as school or college.
- This sentence is possible. It states that definite articles (like "the") are not used before public institutions such as school or college.

9) You must be careful! You are going to fall.
- This sentence is possible. It expresses caution to the person by saying they must be careful because they are going to fall.

10) Do you say "tuxedos" to indicate the plural?
- This sentence is possible. It asks if "tuxedos" is the correct word to indicate plural form of "tuxedo".

11) Do you say "a long-sleeved T-shirt" or a "long-sleeved shirt"?
- This sentence is possible. It asks whether to use the term "long-sleeved T-shirt" or "long-sleeved shirt" to describe a specific type of clothing.

12) A long-sleeved T-shirt or a T-shirt with long sleeves.
- This sentence is possible. It provides two options to describe a T-shirt with long sleeves.

13) A T-shirt without sleeves or a sleeveless T-shirt.
- This sentence is possible. It provides two options to describe a T-shirt that doesn't have sleeves.