Two questions, two sections to an answer. I have some confusion as to how / where the sun disappears to only reappear the next day, but can still work with this.
Climate - One of the best ways to view what is happening here is simple energy. Lets say the Earth receives x energy in the course of a day, or 365x over the course of a year. 2-4 days per month (we will average it to 3 days per month) is 36 days of no light, or about 90% of the total energy from the sun. Now you can make up for it, as 36 days of the year could have a sun with double the energy...otherwise one would expect lower temperatures around the globe.
Ultimately climate/weather can be considered the redistribution of energy across an unevenly heated globe. Thermohaline circulation, also known as 'the great energy conveyor belt' is a process where warm water from the tropics flows along the surface to the polar regions, cools, then descends and travels back to the tropics. Even a globe without a sun for the day will continue to see this circulation continue...it will keep temperatures somewhat consistent around the globe (IE, tropics cool at same rate as polar regions during the 'no sun' days).
Back to the weather statement...if weather is the redistribution of energy across a unevenly heated globe, than increasing the amount of uneven heating (a day with sun vs a day without) you are going to significantly increase the forces that are redistributing this energy. I would expect much higher winds on this planet, and not just on the sunless days.
Plant life is easily adaptable I would beleive...there would be a need to store and increased amount of energy for the longer days and these plants would likely be forced to find a way of saving for the sunless day. Most likely this will result in more advanced root systems that can be used to store the food. animals that eat roots (boars for example) would likely have a bit of an edge in locating food. The advanced root system would also help vs the winds and erosion that these sunless days would cause.
Multiple sunless days in a row would be a more extreme event, but would not be catastrophic by any means. The temperatures would dip significantly, but creatures evolving in this world would have adaptations to help them survive these events.
Culture - The sun is at the core of much of religion as the light/heat/energy bringing being from above. A disappearing sun would be heavily reflected in these religions, likely as a sign/omen (I'd actually think a different sun everyday would drive a 'red sun = glory in battle', 'orange sun good for planting', etc...). A disappearing sun could easily represent a gods wrath blocking the energy lifeblood to this planet.
This is more speculation...but part of Stonehenge appears to be rooted in the sun, in particular the winter solstice where the sun disappears for the longest time in a year. I've seen a TV special that speculated there would be rituals around calling back the sun for the next year and the next rotation. If the sun disappeared for a day, you would expect some sort of ritual 'pacification' of the gods to ensure they returned the sun the next day.
I would suggest Astronomy would be a much further explored science...much of our limitations in viewing the starscape comes from the sun as it interferes with our ability to see the starry sky. With no sun (and therefore no moonlight), the sunless star scape would be impressive and I would speculate beliefs around the stars coming out to inspect earth during these days.
Important other considerations:
Gravity. No sun = no gravity = nothing to pull on earth, letting it drift in one way or another. This would cause an increasingly unstable orbit and may severely increase the level of tilt the planet has increasing seasonal effects. I doubt you could get anything close to a stable orbit during this and there will be very much a lack of consistency on this world. Farmers would have far more of a guessing game as to when to plant crops (a giant belief structure around predicting when to plant would likely come about) and it's questionable how self sustaining farming really could be. You can handwave all this away, but there is a certain level of disbelief that this setup could result in anything resembling stability.
Tides. Same thing as above, but the tidal components of this globe would be horribly unpredictable, even more so if it lacks moons.