PCF8575 woes (driving a ULN2803)
In answer to your third question it looks like the TPL7407L is a pin compatible version of the ULN2003 with MOSFET drivers and negligible input current - it could be driven directly from the PCF8575 without pull-ups.

As commented by Peter there is a 1 Megohm pull-down on the inputs but the PCF8575 has a 100uA pull-up so it should drive it ok.  Designing from scratch I would put pull-ups to give more drive current but it should be acceptable to use in your existing PCB.

tpl7407l Datasheet