Yes they can. Will affect yolk color in sufficient quantity. Negligible affect on nutrition of the eggs.
Helps not at all with parasites. **May** help deter rodents from eating feed if its mixed in.
Metal storage containers work much better. Research re: laying is a bit inconclusive - but every reliable, scale study I've seen says it absolutely doesn't, and those who have the most financial incentive to make use of it if it did help (i.e. Tyson, etc) don't. That tells me that any effect is so tiny as to not be worth the cost or the effort - or they would be doing it already. Effort is negligible, they could have it added to the feed as part of the normal mix, so that's not why they avoid it. Cost? Bulk red pepper runs $650 to 800/ton SO that's probably not it either, if the difference in rate of lay was significant.
As always, don't believe everything you see on the internet, particularly re: chickens.