I'm thinking the $1,500 must be for the projector & screen & mount, not for everything, as there is already over $2,000 of equipment in the list given.
NOTE: None of the links you provided go to the specific products. Seems like a copy/paste from another website or something so I can't look at the details on anything.
Ease of projection is kind of a silly term. Features such as lens shift, and long zoom ranges cost more, and are realistically outside of your budget, so you aren't going to get ease of installation with this. What you can get is a good performing projector for not a lot of cash, and then you have to be very careful with your setup to ensure you get everything correct.
You really didn't describe the basement.
Room with no windows.
Seriously? How is that different from a room WITH windows after dark?
Ceiling height?
How wide?
How deep?
What color paint?
Are you prepared to paint?
What color carpet?
What color ceilings?
Are you prepared for diminished quality if the answer is 'white' or 'light colored' to any of those questions?
How important is the overall quality?
3D is basically included on all new projectors, so just don't worry about whether or not a projector has or doesn't have that feature. Most models handle it reasonably well, even the cheap ones, but almost none come with glasses which would raise the cost.
What is your seating distance?
What screen size, in diagonal inches, is desired?
Do you have flexibility with projector placement?
The only number you mention is 6-8 feet. I have NO idea what that means. Is that your desired screen size or projection distance or something else?
Let's call your goal a 110" diagonal image viewed from about 12' away.
I would be considering the BenQ W1070, W1080ST, or the Optoma 131x on the cheap side of things. They all look good and are all priced under $1,000.
Add a Chief RPMAU (or similar Chief Elite) mount from eBay for under $100.
ie:
Chief RPM000 50lbs Max Capacity Black Ceiling Mount | eBay
Oh, forget that!
But this right now:
Chief InFocus Universal Projector Mount 25lbs Limit MSP Ifrsmaus New | eBay
If you don't buy it right now, then call your friend and have him buy it. This mount is exactly the same as this one:
Chief Rsmaus Mini Elite Universal Projector Mount | eBay
So, at under $50, it's a steal and will make overall setup much easier.
Now, for a 110" screen, I love the Carada fixed frame screens and you should always use a fixed frame screen if possible. But, their 110" fixed frame Criterion screen in Brilliant White is $900 which may be more than you want to spend.
So, you can pick this up for $300 shipped to your door:
Amazon.com: Elite Screens ER110WH1 SableFrame Fixed Projection Screen (110" Diag. 16:9, 54"Hx96"W): Electronics
That's about $900 for the projector, $50 for a mount, $300 for the screen, and you will need some accessories for the mount, call it $30 more.
For cabling to the projector, unless this is an EASY run, you will want to run at least 3 pieces of cat5/6 cable along with at least one 22 AWG HDMI cable and power.
Speaker wiring should be 14 gauge minimum.
Lighting is another discussion altogether and is one of the most important considerations for a multipurpose room IMO.