We are not in possession of the first hadith that were written. I have not come across a text specifying which statement of the Prophet (S) was first written down during the time of the Prophet (S) ; however, Imam 'Ali (A) used to write down the teachings of the Prophet (S) regularly and lengthily, and this became what was known as Kitab 'Ali (A).
Most of the hadith that were written down in the early period of Islam were lost/destroyed; however, an early manuscript has been found (you can look into Sahifah Hammam ibn Munabbih). An early extant collection of hadith is the Shi'i collection Kitab Sulaym ibn Qays.
As for determining which hadith are true or false, you can look into books on the hadith sciences, and as a general rule, correct hadith should agree with the Qur'an and what is generally known about history and the world.