Hello, Timote . Well, a very interesting question. I am sure there are many gardeners here who can advise better than me. But since this is the first reply, I shall try to help. I don�t know why the seeding failed. This is usually a good, but very time consuming way of laying a lawn. Turfing can be expensive, since you are basically paying somebody to grow the turf, and then sell it to you! Grass usually grows anywhere. It could be the weeds are overgrowing your seeding. Using an �industrial strength� weed killer could be an option. Personally, I would not go that way. Chemicals- uurrrgghh! I think, and I will bow to superior knowledge that you need to remove the weeds. Then seed, try to keep the birds from eating it. Not easy, but I believe a good technique is to draw a large eye on paper, and pin it to the ground. Seed eating birds then think this is a predator, and tend to stay away. Another technique I have heard of is to mix a handful or two of grass seed in a bucket full of compost and leave it in a warm, dark place until you see signs of growth, then scatter the contents of the bucket over the area you wish to grass. Birds will not eat germinating seed, apparently. As for the tree, I do not think this is a major problem, since your post suggests you are not having success anywhere. However, to answer your question there are grass seeds (originally Australian, I think), that are bred to grow in shaded conditions, such as under a tree. They are a little more expensive, but worth the cost if it is you only option.
To address the time for laying a lawn, then I suggest early in the year, after the last frost or late in the year, before the first frosts. So March/April or September/October. The main three things that a seed needs to germinate and a plant needs to grow are light, warmth and moisture. I wish you good luck and if you are successful, post back so that others might enjoy your new found knowledge.