What is the proper name for the Buddha's earliest followers such as Ananda and Sariputta - the ones that followed the Buddha during his lifetime. Would the term be bhikkhu and bhikkhuni or does this only apply to settled monastics?
The earliest records we have refer to Ānanda in many ways. But bhikkhu is one of them. For example in the Mahāpadāna Sutta we find
Mayhaṃ, bhikkhave, etarahi ānando nāma bhikkhu upaṭṭhāko ahosi aggupaṭṭhāko.
A bhikkhu called Ānanda, monks, was our servant, the best of servants.
Similarly in the Kalyaṇamitta Sutta (SN i.87)
Atha kho, mahārāja, ānando bhikkhu yenāhaṃ tenupasaṅkami
Then, your Majesty, I approached the bhikkhu Ānanda...
However he is frequently referred to as āyasmā ānando roughly Elder Ānanda. His ordination as a bhikkhu was probably fairly conventional. It's not recounted in full. But at least twice his preceptor (upajjhāya) is mentioned, e.g.
Tena kho pana samayena āyasmato ānandassa upajjhāyo āyasmā belaṭṭhasīso araññe viharati. (Vin 4.86) At that time Belaṭṭhaṣīso, the preceptor of Ānanda, was living in the wilderness.
Sāriputta's ordination was a bit less convention. He was converted by Assaji and already a stream-entrant when he met the Buddha. Curiously I cannot find any evidence of Sāriputta being called a bhikkhu in Pāḷi.
In some discourses that seem to be early the Buddha ordains men by saying ehi bhikkhu "come monk", but this was soon replaced by the more formal two stage ordination process.
On the whole I think bhikkhu and bhikkhunī are fine. They really refer to a lifestyle rather than being an ecclesiastical title.
In Maha-parinibbana Sutta, for example, they're referred to as "bhikkhus" and "Venerable".
In this translation it's "monks" and "Venerable".
Wikipedia says that,
Bhikkhu literally means "beggar" or "one who lives by alms".
The 'ordained' monastics and the Vinaya existed (were created) during the Buddha's lifetime.