Is landlord responsible for my damages in injury that was caused by slip and fall due to snow and ice and walkway

Premises liability questions basically boil down to the issue of, did the landlord (or managing agent if the landlord is not on the premises or subcontractor hired to clear the snow) have notice of the defective condition and enough time to correct it? (In your case snow).

If the condition is created by freshly fallen snow, the responsible party has a reasonable time after the snowfall ended to clear snow. What is a reasonable time? Well, the snow and/or icy condition should have existed long enough for the responsible party to have known about it and had the time to correct it -- certainly a few hours during the day would be enough time. Also, if the responsible party had cleared snow but made the condtion worse, there may also be liability.

Another issue is, to what extent you increased your own risk by taking another route to your car or did the landlord's failure to remove snow force you to take an alternate and hazardous route to your car?

You have three years from the day of the accident to bring a premises liability negligence lawsuit in New York. You should contact the local bar associaton to find a good personal injury attorney in your area.

It depends on a lot of facts that are not in the post. Don't let the statute of limitations run see a Utah lawyer.