Who is Kevat in ramayan, who helped Lord Rama and sita to cross a river?
तत्र राजा गुहो नाम रामस्य आत्म समः सखा | निषाद जात्यो बलवान् स्थपतिः च इति विश्रुतः || २-५०-३३
There, a king named Guha was Rama's friend dear to him as his own
life. He was Nishada by birth, a strong man and well known as a ruler
of Nishadas.
In Ramcharitmanas, Guha is called as Kevata at few places.
When and which river they meet?
विष्णुपादच्युताम् दिव्यामपापाम् पापनाशिनीम् | ताम् शङ्करजटाजूटाद्भ्रष्टाम् सागरतेजसा || २-५०-२५
समुद्रमहीषीम् गङ्गाम् सारसक्रौञ्चनादिताम् | आससाद महाबाहुः शृङ्गिबेरपुरम् प्रति || २-५०-२६
Rama, the mighty armed, reached the river Ganga, which is devoid of
sins and which dispels all sins, which had fallen from the mass of
matted hair of Lord Shankara through the spiritual power of Emperor
Bhagiratha, which is rendered noisy by cranes and herons, which is a
consort of the ocean and which is in the vicinity of
Sringaverapura(the modern Singraur).
This location is confirmed in Ramcharitmanas too
सीता सचिव सहित दोउ भाई। सृंगबेरपुर पहुँचे जाई।।
After crossing the river, When Lord Rama praised him giving a ring, then how did Kevat come to a conclusion that "both of their jobs are same, so how can I take something from you?"
Your question is not clear, this is what Ramcharitmanas says:
पद कमल धोइ चढ़ाइ नाव न नाथ उतराई चहौं।
कहेउ कृपाल लेहि उतराई। केवट चरन गहे अकुलाई।।
नाथ आजु मैं काह न पावा। मिटे दोष दुख दारिद दावा।।
बहुत काल मैं कीन्हि मजूरी। आजु दीन्ह बिधि बनि भलि भूरी।।
अब कछु नाथ न चाहिअ मोरें। दीनदयाल अनुग्रह तोरें।।
फिरती बार मोहि जे देबा। सो प्रसादु मैं सिर धरि लेबा।।
बहुत कीन्ह प्रभु लखन सियँ नहिं कछु केवटु लेइ।
बिदा कीन्ह करुनायतन भगति बिमल बरु देइ।।
As per Valmiki Ramayana, Guha himself didn't go in boat and commanded his kinsfolk to row them (Rama, Lakshmana and Sita) across the river. So not sure from where you got this question.
How did he assume that Lord Rama is an Avatar of Vishnu?
As described in above shloka he was Rama's friend and he was aware of:
मागी नाव न केवटु आना। कहइ तुम्हार मरमु मैं जाना।।
चरन कमल रज कहुँ सबु कहई। मानुष करनि मूरि कछु अहई।।
Why Kevata washed Lord Rama's feet?
First of all it was tradition to wash guest's feet. Another reason which is found in Ramcharitmanas is as follows:
छुअत सिला भइ नारि सुहाई। पाहन तें न काठ कठिनाई।।
तरनिउ मुनि घरिनि होइ जाई। बाट परइ मोरि नाव उड़ाई।।
एहिं प्रतिपालउँ सबु परिवारू। नहिं जानउँ कछु अउर कबारू।।
जौ प्रभु पार अवसि गा चहहू। मोहि पद पदुम पखारन कहहू।।
कृपासिंधु बोले मुसुकाई। सोइ करु जेंहि तव नाव न जाई।।
वेगि आनु जल पाय पखारू। होत बिलंबु उतारहि पारू।।
