Electronic – USB charging and powering circuit with MAX1555 and MAX1881

I think the answer to that question is "yes": e.g. Panasonic - Nickel Metal Hydride Batteries - Technical Handbook 2000 explicitly says that you can't trickle charge NiMH, except for a defined period of a few hours after rapid charging. With NiMH you need to carefully measure the voltage while charging to look for the "end of charge" condition. Having a circuit run directly off the battery will also make that harder.

You could however do:

• on plugging in, start charge
• stop charging as normal (voltage dip detection) and set a timer
• after approx 24 hours, retrigger the charging cycle

Or select a different battery chemistry which can be float charged such as NiCad.

You can't really trickle charge a cell phone through the microUSB interface that most use. There is a minimum current capability, below which your phone just won't charge. A 500 mAh charger will top of your phone, then your phone will stop charging and just draw running current from the USB line.

You are needing a 3-5W solar cell to do minimal charging, plus any losses in regulation. You best bet would be to use a 5V buck converter to drop the voltage of a 12V cell. You are still looking at a solar cell around 250x180mm. This isn't very portable for using while out and about. And full power is only good when angled directly into full sun. Ambient light will give a much lower output.