Here's a thought experiment: suppose I'm in a large box in space without windows or sensors, and I fire a laser in 6 different directions and measure the redshift along each direction. Could that data could then be used to determine my absolute velocity? Note that I'm talking about absolute velocity here, not velocity with respect to anything.
A red or blue shift is created when the light source is moving relative to the detector. In your thought experiment, you emit light and you receive it, so there is no red or blue shift. For your idea to work, light must be emitted not by you, but by the universe equally in all directions. Such emission is known as Cosmic Microwave Background. By measuring its redshift in different directions you can find that we are moving with the speed of 368 kilometers per second toward the Great Attractor. (CMBR dipole anisotropy).
You can obtain a similar result by measuring the average redshift of distant stars.
Also, hypothetically, in a closed non expanding universe, it would be possible to define your speed relative to the universe without CMB or starlight, but by measuring the time for light to make a trip around the universe. However, it would take a very long time.
Your proposal is almost exactly the Michelson-Morley experiment, done in 1887.
It compared the speed of light in perpendicular directions, in an attempt to detect the relative motion of matter through the stationary luminiferous aether ("aether wind"). The result was negative.
No, you can't. If no information can leak through the walls of the box you can only determine your velocity / position / &c with respect to the box.
You can determine if the box is rotating or accelerating, and if it is large enough (or you have good enough instruments) you can determine if it is in a gravitational field.
No. Assuming you're traveling at a constant speed, your box is an inertial frame of reference. By definition, you cannot determine whether an inertial frame of reference is stationary or moving - let alone the speed of movement.
To my understanding, in most cases red shift occurs because the light source is moving relative to the observer - Doppler effect.
 Think of your example in terms of Doppler shift of the train horn. For simplicity, let's assume the train is moving at the constant speed. You will notice the change in horn pitch if you're standing on the platform as the train passes you by. The passenger on train, on the other hand, won't hear it. This is because external observer is hearing the sound waves from a different inertial frame of reference. The passenger is, however, in the same inertial frame as the source of sound waves, the horn.
there is no absolute velocity compared to the universe as you say. You can only have velocity relative to an object (except in a closed universe as per the comment)
EM waves become redshifted, because when they travel towards us from a far away receding galaxy, they will eventually be traveling through a region of space where there is no matter (between galaxy clusters), and in that region of space, space itself is expanding. When the EM wave travels through expanding space, the wavelength of the wave gets stretched, and the frequency becomes smaller, the wave gets redshifted.
your only option would be (since you don't want to measure your speed compared to the box) to compare your speed relative to the EM waves you are emitting. But the problem with EM waves is exactly that no matter your own speed, if you are emitting EM waves, they will still seem to be traveling with speed c relative to you.
Now you could try to release beacons, then try to use EM waves to measure the distance between beacons and the time elapsed between their release, and divide length with time and get your past speed, and assume you are not accelerating, and so get your constant speed, but that could only work if you were traveling with small speeds and not close to the speed of light. Of course in this case you would not measure your speed relative to the universe but to the actual beacons, and even then you could only tell acceleration and not speed. You could set up the beacons so that you have 0 acceleration and constant velocity. But still you would not know your velocity because there is nothing to measure it relative to. And then there is the case about the problem with time. Your time seems normal to you. But relative to what? You could use photon clocks. But even then, distance and time (you use those to calculate speed) are relative to other places in the universe where there are different gravitational fields.