Would you please tell me if the following sentences have been correctly translated:
 Che vestito mi metto stasera?
What dress shall I wear tonight?
 Mi metto i jeans.
I wear jeans.
Can you please tell me if I am correct in thinking that “mettersi” is a reflexive verb that, as shown in the above examples, means “to wear”.
Can you also please verify that the present indicative of the reflexive verb “mettersi” can be translated using the English modal verb “shall” as shown in sentence 1.
the sentences have been correctly translated.
As for the verb “mettersi” in “Che vestito mi metto stasera?” and “ Mi metto i jeans”, it means “to wear” and is a pronominal verb that we call “riflessivo apparente” (literally, "apparent reflexive"), as it seems to be a reflexive verb, but it is not definitely reflexive.
In fact, an Italian verb is reflexive when the action carried out by the subject is performed on the same subject.
For example, in “Io mi lavo” the action of “lavarsi” applies to oneself as e.g. “Io mi lavo” stands for “Io lavo me stesso” (I wash myself).
In short, you can understand whether a verb is a true reflexive or not, when you can replace the pronominal particle “mi”, “ti”, etc. with the direct object pronouns “me stesso”, “te stesso”,etc.
For example, “pettinarsi” [“pettinare se stesso” (to comb oneself)]; “lavarsi” [“lavare se stesso” (to wash oneself)], etc.
See : "Io mi lavo/mi pettino" (I wash /I comb myself),"Tu ti lavi/ti pettini" (you wash/you comb yourself), "Lei si lava/ si pettina (she washes/she combs herself), etc.
Otherwise, the verb is not a true reflexive verb, just as for example "mettersi" in "Che vestito mi metto stasera?" where you could not say " metto me stesso" instead of "mi metto".
Lastly, the present indicative of the reflexive verb “mettersi” can be translated using the English modal verb “shall” as shown in sentence 1.