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I have lived in Hungary for over 5 years and sure I can give you some advise.
Yes, there are English speaking banks where you can open your bank account (savings or salary) account just by showing your passport. They also have internet banking in English. Some of the good banks I would recommend to foreigners are
They also have a customer service call center in English. Besides these, OTP bank, Budapest bank is used by many local Hungarians, but they don't speak English (only Hungarian).
Yes, you will probably need a mobile number to use for internet banking to authenticate transactions as they will send a one time password. It will cost about HUF 1500 for a local prepaid sim from T-mobile or Vodafone.
Be aware that there are monthly charges charged by banks. Some banks such as Raiffeisen require you to just do 3 transactions every month with no monthly charges.
Hope this helps!