There are currently 5,838 surviving full and fragmentary cataloged Greek manuscripts of the New Testament dating back to the early 2nd century (and possibly late 1st century) AD. This was the original language of all 27 books, except possibly Matthew (according to Papias in the early 2nd century AD, who spoke of Hebrew of Aramaic). Around 77% of the entire New Testament is preserved within the first 150 years from the time of composition, and papyrus fragments exist within the lifetime of at least one original author (the Apostle John) or his disciples and long before the Council of Nicaea and Constantine. Archaeologists and antiquities dealers augment the count of cataloged manuscripts annually.
There are also over 20,000 surviving manuscripts in Latin, Syriac, Coptic, Armenian, Arabic, Gothic, Hebrew/Aramaic, and numerous other ancient languages, which date back nearly as far. These do not serve as direct translations into modern languages and dialects, but they can serve as references to the translations from Greek.
Beyond that, there are patristic manuscripts dating to the 1st century AD that reference the original Greek text and cover every sentence of the New Testament many times over. There are also dozens of extra-Biblical texts (some of which are the original documents or carvings) which reference events, people, and places, found in the manuscripts.
Excluding the non-Greek writings, there are an estimated 400,000 textual variants across the nearly 6,000 manuscripts. Since there are about 160,000 words in the New Testament, this seems like a great number, on the surface.
The majority of these textual variants are spelling differences and have no meaning at the word-level of the text or Greek-specific grammatical cases that have no meaning on the sentence-level of the text. These account for about 75% of discrepancies between manuscripts.
The almost 25% of variants come from word differences, such as "Jesus Christ" vs. "Christ Jesus", or easily distinguishable ambiguities, such as in English "an apple" vs. "a napple", that cause no confusion with the text.
This leaves less than 1% of all textual variants that can potentially lead to ambiguities of the original text, such as "666" vs. "616, additions from margin notes, etc. More specific estimates place the count at 0.35%, or about 1400 total. Of these variants (which can include whole sentences), their own wording may be ambiguous, but none of them affect undisputed doctrine in other books that make up the New Testament, including the Trinity, Crucifixion, and Resurrection.
The Evangelical stance of the New Testament is that it contains the inspired word of God, written by the hands of men. The study of textual criticism shows that the doctrine of the 8 authors has remained unchanged for nearly 2000 years, which is inline with that belief.