More specifically to college football or even basketball, is a player who is redshirted allowed to play at all during their redshirt season?
Being redshirted doesn't commit the player to not playing the entire season; the team/player can change their decision part way through the year, but that revokes the benefit of having been redshirted (ie, that year would now count against their eligibility).
If the player plays in at least one game, that year counts towards their eligibility limits.
There is an exception if you start a season as an active player, and then get injured shortly into the season. You can retroactively be classified as a redshirt for that season.
This can vary a lot between sports so I'm not sure if there's a clear answer, but for the most part, once you've chosen to redshirt, you don't play anymore, unless you want to forfeit the redshirt.
In outdoor track, for example, people can red-shirt even if they've already run up to two competitions, and as long as it's before a certain date, and they have an injury.