All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ʻalayhi wa sallam, is His slave and Messenger.
The apparent indication of your questions about the definition of eemaan (faith) and whether actions are included in it reflect your confusion; it may be the result of waswaas (obsessive whisperings) that have overwhelmed your heart and mind.
Our advice to you is to repel and disregard such whispers and pay absolutely no attention to them. If you are confused about certain misconceptions raised by the advocates of religious innovations in this regard, then you should know that fataawa will not be of real benefit to you in terms of eliminating doubts from your heart. Rather, you need to consult with a reliable versed scholar to refute these misconceptions and eliminate your doubts.
In general, the Companions and Tabiʻoon (followers of the Companions) unanimously agreed that actions are a part of eemaan. Imaam Ash-Shaafiʻi said, “The consensus of the Companions and Tabiʻoon and the scholars who succeeded them was that eemaan is made up of three elements: words, deeds, and intention; one single element of the three is not sufficient in place of the other...” This statement was cited on the authority of Al-Laalakaa’i in his book Sharh Usool Iʻtiqaad Ahl As-Sunnah.
Imam Al-Baghawi wrote, “The Companions and Tabiʻoon and their successors unanimously agreed that action is an essential part of eemaan ... They said that eemaan consists of word, deeds, and conviction...” [Sharh As-Sunnah]
You should know that the failure to do the actions that fall under the concept of eemaan does not entail accusing the Muslim of kufr, contrary to what your question implies. The scholars who deemed forsaking the prayer an act of kufr, taking the doer out of the fold of Islam, relied on the relevant ahaadeeth reported to this effect in particular.
Allaah knows best.