Learn Turkish: Ben yalan söylemiyorum.

The verb root is söyle- (to say or tell). To make it negative in the present continuous tense, we add a negative suffix before -yor. Because the root ends in 'e' (söyle-), the negative suffix narrows to -mi to fit vowel harmony. Then we add -yor and the personal ending -um, giving us söylemiyorum.

Why are there two words (yalan söylemiyorum) for "lying"?

In English, "to lie" is a single verb. In Turkish, the action is expressed as a phrase: yalan söylemek, which literally translates to "to tell a lie". yalan means "lie" and söylemek means "to tell". So you are literally saying "I am not telling a lie."

Is the word Ben (I) required here?

No, it is completely optional. The -um ending on söylemiyorum already shows that the subject is "I". Including Ben just adds a bit of emphasis, as if you are saying "I, personally, am not lying."

Why is the personal suffix -um and not -im, given the 'e' and 'i' vowels earlier in the word?

Personal suffixes always look at the vowel immediately before them to decide their shape. Because the present continuous suffix is always -yor, the vowel right before the personal suffix is always 'o'. Under Turkish vowel harmony rules, 'o' is always followed by 'u', so the "I" ending is always -um after -yor, no matter what the rest of the verb looks like.