1. What prayer have to be said during Shachrit and which may one "leave" to be able to pray Shachrit when one hasn’t got enough time for whole, full one (for example when travelling, during brake at work etc.) ? 2. How to deal with situation of lack of time or possibility to pray during proper Zman?

1. If one is late to shul, his main goal is to say the Amidah with the Minyan, for that you can say the minimum following: Brachot Baruch Sheamar Ashrei Yishtabach Shema and Brachot Amidah This can happen once or twice; this is not a regular practice. If you are so late that even if you will say the above you will not make it on time with the Minyan for Amidah – say the whole Tefilah without leaving anything out, ALL of Tefilah is important. 2. This is an educational issue. Tefilah is too important to miss out because of time; make time for Tefilah.