I use a hose level. To make one obtain at least six feet (2 m) of 5/8 inch clear flexible tubing which has an outside diameter of 3/4 inch which is exactly the inner diameter of a standard garden hose. Cut into two lengths of 2–3 feet and slide the tubing inside each end of the hose 4–6 inches. It should form a good enough seal without any additional fuss. (If it doesn't, a wrap or two of Teflon, duct, or electrical tape should be enough.)
Lay out the hose assembly on the patio with each end held vertically with a stake into the ground (or chair, etc.) so it will hold water. (I find it convenient to use one inch brads through the tubing into the stake, but tape or string would work too.) Fill it full so the water level is about half way into the clear ends. Now you can easily determine the same level with extreme precision—perhaps 1/32 inch/0.8 mm at distances apart of 50+ ft depending on the length of the hose. Such accuracy is challenging (at best) with string and a carpenter's level.
To use it, place one end at a reference level and the other at a point in question. Raise or lower either end to cause the water height to increase or decrease, respectively, as doing so decreases or increases—respectively—the volume of the hose level. Once the reference end water level exactly matches the target height, mark the other end where the water is. Be sure to be consistent about using the top edge of the water or the bottom of the meniscus. It is entirely possible the hose has weeped a little water or it has relaxed or tightened due to temperature, sunlight, external moisture evaporation, etc. so recheck the reference end and re-level both ends if necessary and re-mark.
Arrange reference markers (such as more stakes) firmly into the ground at the corners and maybe some points along the sides of the patio and chose a reference height. It doesn't need to be at patio level. In fact, it is better to have it 12–24 inches above the patio level. Mark all the stakes at the reference level using the hose level.
From the reference marks all around, you can measure down to the actual height of the concrete pad or the cement forms. Do whatever slope calculation you desire, like 2.5 inches per 120 inches (a standard recommendation corresponding to 1/4 inch per foot) by measuring the distance from the house to the stake and applying the slope.
For example, if the marker stakes along one side are at 5', 10', and 15' from the house, the slope for each would be
5': (60 inches) is 2.5 / 120 * 60 = 1.25 inch drop
10': (120 inches) is 2.5 / 120 * 120 = 2.5 inch drop
15': (180 inches) is 2.5 / 120 * 180 = 3.75 inch drop
If the reference mark is 17.5 inches above the ground, then the final target marks should be at
5': 1.25" lower than 17.5" or 18.75 inches below the reference mark
10': 2.5" should be 2.5+17.5" or 20 inches below ref mark
15': 3.75" should be 2.5+17.5" or 21.25 inches below ref mark
It sounds like you already have a couple of experts working with you. Without seeing the proposed new space, I can only speak to a couple of your concerns.
There is no easy way to install a sub slab vapor barrier without removing the old slab. You do not want to sandwich plastic between the old and new concrete if you are going to only put in an inch or so of concrete to level the space. I'm sure your contractor will prep the old slab by cleaning etc to assure a good bond with the new concrete. One or two inches of new concrete should be fine, especially in a space with low, lightweight traffic.
I see no reason for a drain in this area unless excessive leakage or standing water is a potential problem.
If radon mitigation is required, the collection pipes should be installed under the existing slab via cutting trenches and tied into your venting before you pour the new concrete. Again, your radon contractor who can see the space should be able to advise you properly and make all the connections to your existing system.
Best Answer
Typically when I pour concrete against a house foundation, (never pour concrete against any siding IMHO, it invites trouble later) I have an expansion joint set to the house. It can be cork, 1/2" thick by 4" tall set to the wall with short pins to hold it while pouring. There is a tar or asphalt impregnated expansion joint material too that may hold a straight line better with fewer fasteners.
The 1/8"/ft is a good number as mentioned by Jimmy fix-it. To set the slope, a form is set in place that sets the size (area) and the top level of the concrete surface. The form is also set at the recommended slope. 2X4s are usually used for this, then they are removed after a day or so. You will also need a concrete edger to ease the corner where the 2X forms are.
Usually pro concrete workers would not set anything within the 10 width, they don't need it, but you can set something at the middle if you choose, since an 8' screed board is a killer to pull if you don't have good help. It can either be a short piece of rebar driven in so the top is at the level for the concrete, strike your concrete in that area to the level dictated by the rebar, after the surface is set, drive the rebar further into the concrete so it is well below the surface. It works but it is easy to loose track of this pin while everything else is going on.
The other way to do the same is set a wood stake in the place you wish to use to set the top of the concrete, making sure the stake is tall enough to be well above the concrete finished surface, say anywhere between 6" to 1' above, (it can be more, but shorter would make the removal harder), set a screw or nail at the level of the concrete and when the level of concrete is set to the nail or screw, pull the stake out and plop a small amount of concrete while you are pouring to fill the void left by the removed stake. Yes you will need boots, and get in the middle of it.
When placing the concrete the edges are floated very close to the finished level, all around the edges. The pour will start from the farthest side, and the pour will proceed closer to the truck. Set the level at the house using a hand float to set the level to the expansion joint and work the far side and start going down the long side, striking the concrete off to the finished level with the screed. You will still need to float it perhaps 2 more times after that if you are going with a broom finish.
Get the rest of the concrete down, and if you are still alive, start the floating process. Use a "bull float" to get most of the surface closed up and the grout brought up, this is critical to get this right the first time!!! Setting the level properly with the screed will really make a difference too. One more time with a hand float after an hour or so should make it ready for the broom finish.
On a slab that is 50" long you will need to run control joints or cut them in later at every 10' or so. I think it is recommended to have expansion joint every 20' but it may not be required. Again, control joints would be highly recommended..