Why can't light move electrically charged object[?] ... But then why can't one move charged objects by lighting them?

The polarity of the electromagnetic wave associated with visible light changes polarity hundreds of trillions of times per second. This is fast enough that a charged macroscopic object won't accelerate noticeably before the polarity changes and it starts accelerating back in the other direction.

Electrically charged objects emit photons (by the way, why don't we see them?)

The chemicals in our eyes that respond to light only respond to a narrow band of frequencies (or if you prefer, a narrow band of photon energies). Therefore we only see light in this band, which we call "visible light".

Why can't light move electrically charged object[?] ... But then why can't one move charged objects by lighting them?

The polarity of the electromagnetic wave associated with visible light changes hundreds of trillions of times per second. This is fast enough that a charged macroscopic object won't accelerate noticeably before the polarity changes and it starts accelerating back in the other direction.

Electrically charged objects emit photons (by the way, why don't we see them?)

If you're asking about the virtual photons that are involved with Cuolomb forces, please see Anna's answer.

If you're asking about true photons emitted from objects either because of blackbody radiation or because they hold oscillating charges, the chemicals in our eyes that respond to light only respond to a narrow band of frequencies (or if you prefer, a narrow band of photon energies). Therefore we only see light in this band, which we call "visible light".