Where can I find the ḥadīth about this ummah being illiterate? Is it found in any strong sources, and if so, weren’t there Companions who could read and write in the Prophet’s time (sallallāhu ʿalayhi wa sallam)?
The ḥadīth I believe you are referring to reads: “We are an unlettered nation (ummah). We do not write nor do we calculate.” This ḥadīth can be found in Ṣaḥīḥ Bukhārī (1814), Ṣaḥīḥ Muslim (1080), Sunan Abī Dāwūd (5319), Sunan al-Nasā’ī (2451), Musnad Aḥmad (5017,5137,6041, and 6129), and Muṣannaf Ibn Abī Shaybah (3:85).
The ḥadīth is ṣaḥīḥ, and as is apparent from the above-mentioned citations, is mentioned in authentic sources of ḥadīth with sound, unbroken chains of transmission.
About your last question, Ḥafiẓ Ibn Ḥajar provides a brief and valuable commentary on the narration in Fatḥ al-Bārī that should suffice us here. He states, “No objection can be raised to the ḥadīth that (amongst the ummah at that time) were those who could write and calculate, since literacy was little and rare amongst them. As for “calculation”, what is meant here are astronomical calculations.” [Fatḥ al-Bārī 4:623]
And Allah knows best.