Yes, your cig lighter will power an inverter if your wiring is fine. Expect more wear than usual on your alt. brushes. How much more....is a good question. The inverters have to run their little fan, etc.
I always kept two inverters at the farm to make sure I could run my mother's asthma nebulizer in case the power went out....and they generally run/deliver 400 watts continuous and some inverters are designed to handle 800 watt "spikes" to get devices running.
The 400 watts will be your determining factor...not sure what the speakers take for juice.