I have a masjid like a 5 minute drive(30 minute walk) from me but I can’t hear the adhan from my home. Is it mandatory to pray there every salah? Also what if the imam there starts or finishes the takbir of movement early or late, would it still be permissible to pray behind them?

Ustadh AbdulAziz al-Haqqan

Not mandatory. Don't fall into waswas.