1) Uncle Willy will wear a green necktie before* he goes to
*I don't understand why he would wear a green tie before he goes to work. If he is actually going to wear it at work, then you would say "...when he goes to work".
2) Tom will cut the wood with a saw because he needs it (what does he need? The wood or the saw?)
Saying "needs" and "necessarily" is tautology.
3) Ahmed will slip and fall over in the mud if he runs
You don't "fall over" mud. You would "slip over" or "fall over" in the mud.
4) After he gives his friend an invitation, Tom will visit him tomorrow evening.
This doesn't make sense. In the first half of the sentence, I take "he" to mean Tom. If Tom has invited his friend somewhere, why does Tom then visit his friend?
5) Tom will run after any thief who
thievessteals something from him.
Although "to thieve" is a verb, we don't use it very often. It's better to use "to steal".
6) Molly will vacuum the room because she promised yesterday
tothat she would do thatit.