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sickmint79, google circuit makati (specifically city kart racing), its a newly (well, quite recently compared to those you mentioned) opened track and its within makati (obviously based on the name) and they have go-karts there that you can rent out.. its also an ideal karting track since its really meant for those not unlike what you mentioned since cars and bikes run there too
the karts in the places you mentioned are quite old also.. you can add carmona as well where you can go karting, its a track for go-karts and small bikes
regarding trackdays, 86/brz owners join a racing event which happens once a month but they start around feb/mar, you can watch and ride shotgun, the organizers allow it if its just for time attack where we usually join, we dont join the grid races usually
for normal trackdays, we regularly go during weekends and even during weekdays depends on our availability but at least once a month, somebody from the group goes
there are some owners willing to let you drive their cars but of course you cant expect everybody will be willing hehe
join the open group so you can already post up and introduce yourself and meet the owners