For weed killers, the only organic options I know are vinegar (kills everything and needs to be done in sun) and pulling. I use a Weed Hound to pull weeds.

For fertilizer, I use coffee grounds from Starbucks. A lot of people use various ground grains.

The thistles....are they a general nuisance or or are they more or less what lawn you have.
You certainly don't want tender feet romping on them...

Introduce yourself to "RoundUp"....and read the instructions carefully.
It might be a never-ending job but you certainly have to get rid of the invader.

RoundUp can be painted on the plant. With a 50 cent paint brush, you make up a solution and dip the brush into it and then paint the leaves and stems of the thistle.
Pretty soon the plant is dead. I don't think though, the problem is solved....the thorns are sill there. You have to remove the plant....carefully.

Trees erupting in your ground....the lawn mower will do them in as they sprout...but the neighbor's tree will keep sending them. If they are type that come from air borne seeds, there's not much you can do but keep up lopping them off with the mower.
If they are coming from roots that are invading your lawn, then you have the right to block such roots.
That might entail a bit of hard digging...and placement of a barrier that will turn away the roots.
You will have to dig down possibly as much as 3 feet, six feet...you will have to discover where the roots are, cut them off there and put the barrier in.

If the tree's roots are of major size, you could be undermining the stability of the tree. That's not your fault...you say. It may very well be your responsibility and the law might take a dim view if you were to harm the tree. While invading roots are usually looked at unkindly, still you don't have the right to do harm to a neighbour's tree without due regard.

This is where diplomacy comes in....talk to your neighbour...tell him what his tree is doing and what you would like him....or you...to do.
He might help in putting in such a barrier.

There is a formulae you can use to find out whether you can safely cut the tree's roots. Measure up the trunk
4 1/2'....there measure the diameter of the tree...in inches....and then multipy that figure times 3....
that is the number of feet out from the trunk you can safely cut such roots without harming the tree.
If they are just feeder roots you might see popping up in your lawn, go ahead, cut them as you please...they wont harm the tree.

Thank you for your responses. As for the tree, I took another closer look around and it's actually a tree that's behind our fence that's sending roots into our yard so it's fair game. =) The thistles are a nuisance more than anything. They are under control for the most part, just need to keep at them. I will look into the coffee grounds, I've never hear of that before. Anyways, thanks again!

Used Coffee Grounds(UCG) are what you want. Starbucks gives them away for free. You could use any ground grain including: Soybean meal, Alfalfa meal or pellets, Corn Meal, Corn Gluten Meal and other. I alternate between soybean meal, alfalfa and corn. I use UCG whenever I get them.

As for weeds. Mowing High (4+") and watering deep and infrequently with 1" of water, no more than once a week is the best approach. I have been organic for 4 years now and by following proper cultural practices I only have a few weeds in my lawn

You know there is an Organic Lawn Care Forum as well. You might get more responses over there or just let us know if you have anymore question.