Ok. I'm a native speaker like marie-hélène.
I'm quite surprised to see that marie-hélène only corrected one sentence. So here are my translations :
Do you have to go now? (formal) = Vous devez partir maintenant ?
Do you have to go now? (informal) = Tu dois partir maintenant ?
Tu dois y aller maintenant ? (this one "y aller" is more informal than "partir". For example, "je dois y aller maintenant" is more often said than "je dois partir maintenant" when you speak to a friend. Or you can say "il faut que j'y aille" for "i need to go now".).
Don't worry! (formal) = ne vous inquiétez pas
Every Monday evening = tous les lundi soirs
Maybe you can say "chaque" to translate "every" but it may depend on the sentence's meaning
Each monday evening = chaque lundi soir
How is the climate? (not weather) = comment est le climat ? Comment est l'ambiance ?
You can say both.
I am engaged. (with a girl) = we almost never say "je suis engagé", more "Je suis fiancé" ou "j'ai une petite amie"/"j'ai un petit ami"
"j'ai un copain"/"j'ai une copine"
I am worried. = je suis inquiet ("inquiète" if you're a girl)
One way (Traffic sign) = "sens unique" : rue à sens unique / voie à sens unique / route à sens unique
Where shall we go ? = Qu'est-ce que tu veux faire?/Où est-ce qu'on va?
Will you have tea or coffee? (formal) = Est-ce que vous voulez du thé ou du café ?
You can be more "direct" and say "Vous prendrez du thé ou du café?"
Will you have tea or coffee? (informal) = Tu veux du thé ou du café?
You can say : "what do you want ? tea or coffe ?" -> "tu veux quoi ? du thé ou du café?"
I think this answer take all the best from other's answers.
I hope it will help to reduce confusion...