Nowadays only the Brahmin caste undergoes upanayana.
For example, we are Kshatriya Varna (by birth) but we have not undergone upnayana.
Question- Is anyone who has not undergone the upanayana samskara a Sudra?
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(This answers the question in the title. I think there must be another question for the rest)
The literal meaning of the word 'Dvija' means a twice-born. So at the moment the upanayana is performed, a person becomes a 'Dvija'. This is because the person is considered to be born again at the end of upanayana. Note that even birds are called 'dvija' since they have two births (once as an egg and once from the egg). I don't have the exact source of this but a reference to any sanskrit dictionary will reveal that any oviparous animal or bird is referred to as a dvija. (Example: http://sanskritdictionary.com/?q=dvija)
However, the word 'Brahmana' is used to denote a person who performs the required religious acts and tries to understand 'Brahman'. In the Chandogyopanishad, Uddalaka mentions to Shvetaketu that a person who does not know 'Brahman' is called a 'Brahma Bandhu', or a relative of a 'Brahmin' (in that context), and not a 'Brahmana'.
॥ षष्ठोऽध्यायः ॥ ॥ प्रथमः खण्डः ॥ श्वेतकेतुर्हारुणेय आस तँ ह पितोवाच श्वेतकेतो वस ब्रह्मचर्यं न वै सोम्यास्मत्कुलीनोऽननूच्य ब्रह्मबन्धुरिव भवतीति ॥ ६.१.१॥
Meaning: There was a son of Aruneya (Uddalaka) who was known as Shvetaketu. The father said 'Shvetaketu, live in Brahmacharya just as the rest of our 'kula' else you will be a ब्रह्मबन्धु.
Yes anyone who has undergone the Upanayana Samskara is a Dvija.
But, as per Smritis, only the males belonging to the first three Varnas (i.e Brahmin, Kshatriya and Vaishya) are eligible for the Samskara and hence for Dvijattva.
Dvi+ Ja literally means who are born twice.
For them the first birth is from the mother and the second is through the initiation into the Vedas, through the Gayatri Upadesam.
Manu Smriti 2.169. According to the injunction of the revealed texts the first birth of an Aryan is from (his natural) mother, the second (happens) on the tying of the girdle of Munga grass, and the third on the initiation to (the performance of) a (Srauta) sacrifice.
Manu Smriti 2.170. Among those (three) the birth which is symbolised by the investiture with the girdle of Munga grass, is his birth for the sake of the Veda; they declare that in that (birth) the Sivitri (verse) is his mother and the teacher his father.
As per Smritis, Sudras are not even allowed the Upanayana. So, they are by virtue of their birth, not Dvijas.
Manu Smriti 10.126. A Sudra cannot commit an offence, causing loss of caste (pataka), and he is not worthy to receive the sacraments; he has no right to (fulfil) the sacred law (of the Aryans, yet) there is no prohibition against (his fulfilling certain portions of) the law.
Manu Smriti 10.127. (Sudras) who are desirous to gain merit, and know (their) duty, commit no sin, but gain praise, if they imitate the practice of virtuous men without reciting sacred texts
"Is anyone who has not undergone the upanayana samskara a Sudra?"
By birth alone everyone is just like a Sudra. So, yes. It is only by the purificatory rites that one attains Dvija-hood.
Manu Smriti 2.172. (He who has not been initiated) should not pronounce (any) Vedic text excepting (those required for) the performance of funeral rites, since he is on a level with a Sudra before his birth from the Veda.
Here the Sanskrit words being used are
"Sudrena Hi Samas" which means "same as a Sudra". So, one who does not have Upanayana is the same as a Sudra.
The Skanda Purana also has the following that answers your question:
Janmanā jāyate śūdraḥ saṁskārad bhaved dvijaḥ.
Everyone is born as a sudra and by samskara one upgrades to dvija status.