Here's a few basic generalizations which hold true:
Omnis are considerably less susceptible to wind noise than cardioids. That is partly due to the physical design of the omni capsule which puts the diaphragm under higher tension.
Windscreens of both the foam and fabric-covered-basket type help to reduce or eliminate wind noise, regardless of microphone pattern. The effectiveness of a windscreen is determined by how well it can calm the turbulent airflow around the capsule. This is more or less directly related to its size. Small windscreens don't trap much air-space and are therefore not very effective. The size of the microphone itself has little or nothing to do with it. If you were to put a Church Audio or small AT cardioid in a big Shure A81WS instead of the tiny foam screens typically used on them, you’d be likely to eliminate all wind noise in most cases.
Windscreens reduce the microphone's high frequency sensitivity, but do so in a very linear fashion which can usually be compensated for effectively with simple EQ. Some screens are more eggregious than others, and better screens effect the highs less. However, in general the bigger and more effective the windscreen is, the more it will reduce the mic's high frequency sensitivity.
Furry covers work on the same basic principle of reducing the energy of the wind turbulence around the capsule. They are popular because they can be added to an existing windscreen to increase its effectiveness and because they are inexpensive and compact to carry. They are not any more effective than using a larger foam screen to begin with and in many (most?) cases are less effective than a larger foam screen would be. As far as fur covers go, the looser, fluffier and fuzzier the fur, the better. Matted, dreadlocked fur isn't particularly effective at reducing wind noise, yet is no less effective at rolling off more treble response. If the fur isn’t visibly blowing around freely in the wind, it isn’t doing its intended job.
You can also nest a larger foam screen over your small foam one to increase wind protection. Ubiquitous and cheap ball-mic slip-on foam screens (SM58 style) work well for that to bolster the wind protection of small microphones, velcro-tied or gaff taped closed at the open, cable exit end. They sell those at my local dollar store for a buck. If on a tight budget I’d try that first, then add fur if that wasn’t enough, or go directly to the highly effective but somewhat costly Shure A81WS. That big foam Shure screen is basically two windscreens in one, with denser wind blocking foam on the outside, and more open ‘air-space’ foam on the inside, used primarily to center the microphone in that internal airspace. It’s very effective and well worth the cost, IMO.
I can get away with small lavaliere foam screens on omnis in the same outdoor situations which call for use of the big Shure A81WS on cardioids or hypercards. I don’t care as much for fur covers unless the foam isn’t enough and I don’t have a bigger foam screen to substitute.
The key to effective wind protection is sufficient size of the dead-airspace around the capsule housing.