This is probably the intended solution for part 1. Nice find! For part 2 I have a solution but I don't think it is the intended one... Working on a video.

Congratulations KennKong, this is the intended lifting method!
Good luck to everyone with part two!

Presumably unintended solution for part 2:
h6UsXc-jSgo

Oops, I thought I fixed that already!
Thanks Jonatan, I'll have another small update soon.

Well, we know now that grayarea is a sore loser. We can also deduce that since 3 cubes through the door is not intended (no matter how doable), and that one cube is impossible, let's work on two!

The door remained open again, this time after the first activation. I've seen this with buttons before, where if you can trigger them twice in rapid succession, they remain activated.

Congrats, KennKong! You have my respect...

I'm not beaten yet KennKong! Actually I'm really looking forward to having it done..
Here is another update (lucky 13) which makes the area under the laser even more hostile and should block Jonatan's shortcut.
I also cleaned up the fix for Ruien's solution near the hole in the glass.

KennKong wrote:
The door remained open again, this time after the first activation. I've seen this with buttons before, where if you can trigger them twice in rapid succession, they remain activated.

I'll try to replicate it, but I suspect my laptop isn't fast enough to cause that problem...

For part 2 only :

No Audio Available
uEaB9CrLGEA

http://youtu.be/uEaB9CrLGEA

I don't know how much this overlaps with the intended way.
To be clear, at around 0:08 to 0:09, I pressed E to grab hold of the cube through the grating.

Xtreger, that's damn clever again. Nothing like the intention, so I'll respond to it more later on.
Nice one!

Don't know if it is relevant but another way to remotely get the cube off the grate onto the small platform is to give it sideways momentum by rapidly shooting tractor beam portals and then remove the tractor beam making the cube with some luck land on the platform.

Okay, I solved it, but it might not be the right solution either. It's not worth making a video of since there are probably better solutions out there.

it_works
1.) Do part 1 as in the intended solution discussed earlier.
2.) Grab a cube from the tractor beam and place it on the button. At this point there are two cubes in the funnel, one on the button, one on the floor, and one in the air pointing at the door target
3.) Grab the cube on the floor and float with it to the laser. Aim laser at the the cube in the air through the portal in front of you and activate the door.
4.) Carry this box through the door and throw it in. Go back to funnel.
5.) Grab another cube inside the funnel and use it to direct the laser in the same way. (You can do this with up to 2 cubes but not more than this). Carry this cube through the door. You are now on the other side of the door with 2 cubes, and funnel is active on other side.
6.) Go upstairs and emerge at the top. Move the sky cube as far right as possible.
7.) Go over to the cube and jump off the edge so that you are sucked up by the funnel and end up with the cube.
8.) Grab the cube and place it on the ledge below you. Stand on it.
9.) Move portal to other target (to your right).
10.) Grab cube and jump forward, turn left and use the thin glass wall to slice between the portal gun and cube, so that cube is now on the other side of the glass while you float in the other direction. Cube is now next to the fizzler stack (I've never had it be fizzled in the several times I've done this).
11.) Shoot portals on back wall and use the momentum to get out of this area and back over to the fizzler stack.
12.) profit

It seems like the intended solution is to do something more like drop the cube somehow from its high spot by turning the tractor beam off, and simultaneously placing two portals- one below the sky cube and another on the top platform, so that the cube oscillates through the portals high up, and then you go upstairs and grab it. I haven't found a way to consistently grab the second cube while also having time to turn off the tractor beam and placing the two portals for the sky cube's oscillation all while getting through the door in time. But it can probably be done with some practice.

Jonatan, Ruien and Xtreger: some very nice points! I'll have a new version soon.

jonatan wrote:
Don't know if it is relevant but another way to remotely get the cube off the grate onto the small platform is to give it sideways momentum by rapidly shooting tractor beam portals and then remove the tractor beam making the cube with some luck land on the platform.

I'd already tried that - unfortunately I could not succeed in doing so..

I had also managed to oscillate the cube through 2 floor surfaces LONG before (in the very early versions), as Ruien said. But I couldn't think of a way to proceed after that. I'll try that approach again soon...

So guys I've had to sabotage some more and update to v1.14. I've removed the pesky ledge near the grate and introduced a very nasty corner on the top fizzler.
Again, I'm not trying to erase your clever solutions but to help narrow down your search a little.

Here's a solution that should work for v1.14 as well :

2tEoCqccdM4

http://youtu.be/2tEoCqccdM4

I feel like a complete idiot for not having noticed the door receptor's warm-up behavior properly before.

If that's not the intended solution for the second part, I can't imagine what else he had in mind.

I'm confused by why this works though. I gave up on yoyoing through the laser portal long ago, when he added that little tab to the side. Also, in v1.14, he added a fizzler to that portal (but maybe only at the edge, so you can't stand on a cube?)

I really hope this idea is not contained in the solution. From the very beginning I concluded that it is not enough time for the door to open with a bouncing cube because it needs a warm-up period. I did recently realize just like xtreger that if you hit the receptor twice each bounce cycle it actually does open it eventually. However this is very unintuitive behavior and I dismissed it as unintended.