You or your builder need to familiarize yourself with local building codes to find your answer.

Gnerally, it is against building codes to deliberately divert water from your property to someone else's private property, and that is what it sounds like you want to do. You want to regrade your yard so that it drains into the woods, which you do not own. Just because you are "First come" does not mean that you can divert your standing water (backyard lake) onto that privately owned land.

It sounds like you don't live there yet. Be glad this problem surfaced now, rather than after you moved in. Your builder will be more motivated to fix the problem now than he will be after he has received his final payment from you.

A project is supposed to meet the grades of the adjacent properties at the property line. If the builder owns the land adjacent to yours and is willing to adjust the grade of that lot along your property line, that is OK. When it comes time for him to develop that, he will have to meet the grade that exists on the property line at that time. Sometimes grading easements are granted (or sold) by a land owner to allow an adjacent landowner some relief for grading across a property line. No one is obligated to grant such an easement.

Water is also not supposed to drain off of a site at a greater speed or volume onto another property after development than it did prior to development. This can be hard to prove.

This situation illustrates the importance of having a professional site plan that shows grading and drainage prior to construction. People will avoid the expense and trust their eye or that their builder is going to build something that works. It does not mean that the builder is dumb or is a shady character. It is just not that easy to guess. If an engineer or a well trained professional starts out with a good topographic survey, they can calculate all the grading and drainage to accommodate such things as basement windows, walkouts, and about anything else to KNOW the results before the site is developed. Obviously this was not done, or there were changes (such as adding bigger basement windows) after the plans were drawn. You can look at this as an excessive cost, but I think you'd agree that it would have more than paid for itself in this case.

Builders build houses. Not many are trained site planners.

You can continue with looking to the builder (who has other interests) for solutions, or you could bring in a civil engineer or a landscape architect (only one who is particularly experienced with grading and drainage calculations) to correctly fix the problem. In many cases this can be done with added aesthetic value rather than just taking care of the water issue. It is really not something that can be solved by an internet discussion. The best you can get here is advice on who to contact rather than specific directions on how to fix it.

I hope this helps and I hope others read this and understand the importance of a good site plan prior to construction. Hopefully, some others will not let themselves get in this predicament.