Phone charging current draw
Yes, phones do cap their charging current. The cap depends on "charger signature" provided through D+/D- USB lines from the charger side, and on the state of charge (SOC) of their internal battery, plus consumption of internal CPU/display and current state of their operations (sleep/active etc.).

If you don't provide any signature (leave D+/D- floating), phones will cap their consumption at about 500 mA.

And yes, you do need to "cap" your power delivery in accord with certain I-V curves that are detailed in BC (battery charging) and PD (power delivery) USB specifications, which can be found on usb.org website, in "developers" section.