Learn French: Mon coude touche la table. - My elbow is touching the table.

In French, the present tense covers both the English simple present ('touches') and the present continuous ('is touching'). You do not need a separate verb for 'is' or an '-ing' equivalent here. The context tells us this action is happening right now.

Why is it Mon coude and not Ma coude?

Because coude is a masculine noun. In French, possessive adjectives must agree with the gender of the noun they describe. Many body parts have unpredictable genders, so you just have to memorize that it is un coude (masculine) while others like une jambe (a leg) are feminine.

I thought French used le or la for body parts instead of 'my'. Why does this sentence use Mon coude?

It is true that when you do an action to your own body, you usually use a reflexive verb and a definite article (like Je me lave les mains for 'I wash my hands'). However, when a body part is the actual subject of the sentence doing the action, you use a possessive adjective like Mon just as you would in English.

Does the verb toucher ever need the preposition à after it?

Yes, but it changes the meaning! When toucher is followed directly by an object, as in toucher la table, it simply means 'to physically touch'. If you use toucher à, it usually means 'to meddle with', 'to alter', or 'to tamper with' something.