Importance of Tipitaka in Mahayana Schools

I'm sure many different Mahayana schools do many different things, so it's impossible to generalize, but it is certainly true that, contrary to popular notion, Mahayana tradition does not reject the Pali texts, many of which are found in translation in both the Chinese and Tibetan canons. It is, moreover, a violation of the bodhisattva vow to disparage the Hinayana or dissuade anyone from Hinayana practice. In the Tibetan tradition, the Pali texts are studied quite seriously and the Dalai Lama has been known to cite Theravadan authorities, eg., Buddhaghosa, as authoritative. The issue is not acceptance but interpretation and acceptance of additional non-Pali sutras as "Buddhavacana."

1. Is Tipitaka taught and learnt by monks in monastery?

Yes, traditionally. From the beginning one would learn to cite them. Today of course, many have adopter other ways like simply oral, by reading books and just learning to spell it right.

2. How often Tipitaka is referenced in lectures by monks? (for e.g. I was once listening to discourses of Dalai Lama and he was talking of Nagarjuna, Milarepa, Ashvaghosha etc. It might be that he did not reference Tipitaka in those lectures but might have referenced in some other lectures)

Most would do it as good as possible. Its certainly so that people who have developed different things are not able to cite and well educated people would certainly know if he speaks the words of the Buddha, in words and meaning.

3. Are laypeople encouraged to learn Tipitaka?

It does not look like as if the Buddha wanted laypeople to learn to cite them like monks, but of course he encouraged to learn and know his teachings independent whether one walks the way as a monk or a layperson.

There are some Mahayana Schools who highly respect them (Pali Tipitaka) or needs to respect them (would be a break of the root vows if disrespected!), such as most Tibetan schools and certain schools who totally reject them, having developed there own texts later on.

Since monks in many Mahayana Schools do not really stick to Vinaya (diszipline), they are also not so open to share those texts to lay people.

In argumentation, the old suttas are in fact very important, even for followers of other paths, since they hardly have anything to relay on aside of later developed texts and interpretations out of own opinions.

Tripitaka is taught to and learned by monks in the monasteries and laypeople are encouraged to learn. From a Tibetan viewpoint, the texts H.H. quoted are included within one of the three baskets. Shantideva's Bodhisattvacharyavatara is included in the Sutra Pitaka (although it is not a sutra), Asanga's Mahāyāna Sūtrālamkāra kārikā is included in the Abhidharma Pitaka, Kamalashila's Stages of Meditation belongs to the Abhidharma Pitaka, Nagarjuna's Karika belongs to the Abhidharma Pitaka as well, and so forth.

In the ninth chapter of the Bodhisattva Deeds, Shantideva explains that anything that is included within one of the three baskets was the speech – or at least the intention – of Lord Buddha. His Holiness the Dalaï-Lama highlighted this principle. In Illuminating the Path to Enlightenment, he writes:

As Dromtönpa lay dying, his head rested on the lap of one of his chief disciples, Potowa, and he noticed that Potowa was crying. Then Potowa said, “After you pass away, in whom can we entrust our spiritual well-being? Who can we take as our teacher?” Dromtönpa replied, “Don’t worry. You’ll still have a teacher after I’m gone—the tripitaka, the threefold collection of the teachings of the Buddha. Entrust yourself to the tripitaka; take the tripitaka as your teacher.”

Furthermore, in 'Transcendent Wisdom', His Holiness writes:

The three baskets are (1) Vinaya, dealing with moral discipline; (2) Sutra, dealing with the training in meditative concentration and other facets of the path of awakening; and (3) Abhidharma (sometimes translated as "higher knowledge"), concerning a diversity of topics ranging from psychology to cosmology. For a further explanation see Opening the Eye of New Awareness, pp. 47-52.

We hold that the Tibetan translators showed to which basket the text they were translating belonged, by way of paying particular homage. As His Holiness writes in his commentary to Stages of Meditation:

The intention [of the Tibetan translator paying homage when translating Nagarjuna and so forth] was to cearly indicate to which of the three divisions [the pitakas, or baskets] of the Buddha's teachings any sutra or commentary belongs. [It goes as follows:] 1. Supplication is made to all Buddhas and Bodhisattvas if a text belongs to the Collection of Discourses [i.e. Sutra Pitaka]. 2. And if a text belongs to the Collection of Knowledge [i.e. Abhidharma Pitaka], supplication is made to Manjushri. 3. In order to indicate that a certain text belongs to the Collection of Discipline [I..e Vinaya Pitaka], supplication is made to the Omniscient.