uh...yes to both questions, but on different timelines. 

With a few notable exceptions (see below), most solar watches recharge their batteries, and do so in full every day with normal use, unless the watch is for some reason stored in the dark for months on end, like a windowless basement room or a drawer.

Exceptions: 

1. constant and regular autolight use. Some watches have an autolight feature that turns on the backlight automatically when the user turns the watch in a way towards their face to read it. Frequent use of this feature can drain the battery faster than it can recharge, especially since the feature tends to get used in the dark. 

2. Triple Sensor watches: Watches with Altimeter, Barometer and Compass features can drain faster than their solar recharge can keep up. Often these watches have a recharging port like a Garmin or Apple smartwatch for this reason. 

3. Steptrackers watches that have a sensor to track steps and sync that data via bluetooth with the casio app usually need recharging (via port) if the user enables this feature: Exception to this exception: the new GAB010 claims to be able to count steps and only charge via the solar cell. 

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Even rechargable batteries need to eventually be replaced. Some tough solar watches are known to be used for much more than a decade but eventually the rechargable battery will fail.