I'm a Czech student, who unfortunately has to study English Syntax for one semester (oh man, it's so hard!). We got this homework and I'm not sure if I really understand the task:

1, (i) Underline the Verbs/ Verbal parts of the main clause Predicates

(ii) How many elements does eth 'Verbal part' of each predicate

(iii)Which part of the complex tends to carry grammatical categories of the Verb
(Tense/Aspect, Agreement) and which is the locus of 'lexical meaning'? Answer this in
terms of right/ left side and w.r.t. the categorial status of the relevant element.

(a) Mary also (reads exciting novels)vp.
(b) Johanna might (run away with the sailor)vp.
(c) Mr. Todd had been (shaving the customers with real enthusiasm)vp.
(d) My parents are not (looking forward to his arrival).
(e) Some people (take showers twice a day)vp.

(f) Jane was allowed to (look at those pictures)vp.

I guess I know the Verb phrases or Predicates (I put them in the brackets and wrote "vp" next to them).

The Verbal parts are: reads, run, shaving, looking, take, look

Then in (ii) - how many elements does it have - am I supposed to count the words or the whole phrases? Or what elements in Verbal parts? If the Verbal part is "run", then there's only one element, isn't it?

And then in (iii) - I thought I undestood the question. The auxilary verbs and some suffixes (etc.) tends to carry the grammatical categories. But that last part confuses me - in terms of right/left side? What does that mean?

Also I would like to ask one more question: Why are Verbs central for sentence structure?

I know it's a really long question and I'm sorry for that. Our teacher uses so many different terms for one event in his script that I don't know what is what.
There are many different teaching techniques for English, and different grammatical terms. So use your textbook's (or teacher's) terminology. If you are confused, you can ask questions!

(b) Johanna might (run away with the sailor)vp.

Most authorities would put the modal auxiliary might with the verb to form the verb phrase.

They would not include the objects and adverbs; those are traditionally other parts of the predicate.

The phrasal verbs are also a bit tricky. Some change the meaning of the main verb, and so the particle is an integral part of the verb.

Verb phrase: might run away (auxiliary: might, phrasal verb: run away = escape, consisting of the main verb, run and the particle, away.) Some texts use the term preposition instead of particle.

(c) Mr. Todd had been (shaving the customers with real enthusiasm)vp.

Verb phrase: had been shaving (past perfect progressive tense)

(d) My parents are not (looking forward to his arrival).

Verb phrase: are looking forward to

look forward to = phrasal verb = to anticipate

present progressive tense.

(e) Some people (take showers twice a day)vp.

Verb phrase: take, simple present tense.

(f) Jane was allowed to (look at those pictures)vp.

Verb phrase: was allowed, past tense, passive voice.

This is a special case of the passive form of the verb allow being followed by an infinitive phrase.

infinitive phrase: to look at those pictures

Allow belongs to a class of verbs (catenative verbs) which can be followed by a gerund, infinitive or bare infinitive. Some texts call the infinitive phrase a noun phrase, the object of the main verb.

I'm not sure what your teacher means by right side/left side, but in diagramming sentences (structural decomposition), the highest level is subject (left side) and predicate (right side).

In this mode of diagramming, a vertical line represents the division of subject and predicate:

Thank you very much for your answer. I see it more clearly now.