1. How much water would be required to flood Earth to this level?
1.1. $V=\frac{4}{3}\pi r^3$
1.2. The radius of Earth at the equator is $6378.1$ kilometers. The radius of Earth plus the height of Mt. Everest + flood coverage is $6378.1+8.849=6386.949$ meters.
1.3. To make this easier, I'm going to assume that Earth is a perfectly flat ball. The volume of Earth, plus flood is $1091361395266.3 km^3$ or $1.0913x10^{12} km^3$. The normal Earth's volume is $1086831493929.56 km^3$ or $1.0868x10^12$. The volume of water to be deposited is $4.5x10^9 km^3$. For comparison, the largest body of water on earth is the Caspian Sea is $69,400 km^3$. All water on Earth is only $1.386x10^9$ cubic kilometers km3. We are adding 3.247x the amount of water on earth. Dang.
2. How could the water be deposited to do the least damage or alteration possible (including heating or cooling or altering Earth's rotation or axial tilt) to Earth - aside from flooding it, of course? We could simply drop the ice as a single bolide, but there would be a great deal of cratering and other damage, the avoidance of which is highly desirable.
2.1. Applying that much mass to the atmosphere in such a short period of time will surely alter Earth's rate of spin if applied to same side of Earth. To counteract this, we will need to send in the water asteroids in pairs, one to each side of the planet so that the impact of one asteroid is counteracted by the impact of the other.
2.2. (I don't have the math/science to figure out how much energy all these asteroids will add to the atmosphere on reentry or how much of that energy will be absorbed by the water in the asteroid itself. I definitely don't have the science/math to figure out how all that water will change Earth's albedo. Sorry.)
2.3. Delivery Alternatives
2.3.1. If the asteroids break up in the upper atmosphere and handwave turn into water droplets then they could come down as rain. However, this much water over land will cause significant flooding and erosion and thus fails the "minimal changes to landscape" requirement.
2.3.2. If the asteroids break up over the ocean and come down as rain on the ocean then there's not much immediate erosion. The oceans will just rapidly rise.
2.3.3. Aliens are "kindly" depositing water directly to the oceans without the heat of reentry.
3. Given that the water must be deposited over a period of 40 days, and assuming an effectively equal rate of deposition over the entire surface of the globe during the majority of this period (at least 99% of the 40 days, with at most the initial and terminal 0.5% of this time ramping up and tailing off the rate of deposition respectively), at what rate must the water be deposited, and how would that compare to natural rainfall?
3.1. To keep to the flood time line, we need to deposit $4.5x10^9 km^3$ over 40 days so that's $\frac{1.125x10^8 km^3}{day}$. Annual global rainfall is about $\frac{5.23x10^5 km^3}{year}$ or $\frac{1433km^3}{day}$. Five orders of magnitude more rain will fall per day than usual. If average daily rainfall for the planet is $\frac{.27cm}{day}$ then a five orders increase would be $\frac{~270 meters}{day}$ or $\frac{11.25 meters}{hour}$. Done. Humanity doesn't last two days since half the world's population lives in coastal zones. There's no way to move that many people. Whether killed by weight of the waterfall rain or washed away with by the tsunamis, everybody dies.
4. How could that amount of water be removed over a period of 220 days, again causing minimal damage or alteration to submerged land features. The maximum preparation time for this removal of the excess water is the aforementioned 150 days, though it may be less.
4.1. Lots and lots and lots of energy. Getting water into Earth's gravity well is easy. Getting it out again under a hard-science requirement is incredibly difficult. According to this What If question about getting just all the humans off earth,
> would tax our resources to the limit and possibly destroy the planet.
Lifting 3.247x Earth's original water supply just isn't going to happen without considerable external power sources. Thankfully, the Magratheans know how to do this kind of thing.
4.2. Extraction Alternatives
4.2.1. What if Earth absorbed that much water like a giant sponge? Huge earth quakes while the crust and mantle open up "pores" to make way for all that water to come in. Huge steam explosions when hot magma hits cold water. Making pockets to hold all that steam is going to make for greater instability later on....thus disqualifying this approach from the "change as little as possible".
4.2.2. The aliens who "loaned" us all that water now want it back. Using their "Infinite Energy Drives (TM)" they shuttle the water off the planet in their Big Gulp mega-barges.
5. What would the world look like after 150 to 370 days entirely under water? What species (animal, plant or otherwise) could survive?
5.1. All the land animals that can't grasp onto floating debris are dead. Drowned. Amphibious or aquatic species may survive if they can make it past the initial flooding though non-carrion food and shelter will be difficult or impossible to come by. Crocodilians may survive just fine as they are accustomed to long periods in the water and can eat carrion.
5.1.1. Quick thinking humans may survive by making small rafts or piling into existing ships. Food and water are going to be a really difficult problem as untreated fresh water doesn't last longer than 2 or 3 months. Not many ships carry sufficient food for a year for max crew and passenger loads. Starvation is the likely end for those humans who make it onto ships.
5.2. Most sea life would die too. Corral reefs host an incredible amount and variety of life, all of which depends on being in relatively shallow, sunny waters. With the addition of 8 km of water overhead, the increase in pressure plus the lack of sunlight would kill practically all coral reef life. Ocean vent life may survive the increase in pressure though I have no way to prove that.
5.3 Vast majority of plant life is dead too. Few planet species can survive long term immersion in salt or fresh water. And those that can, how well do they do under 6 to 8km of water and absolute darkness? Probably not well enough to survive a year down there. For comparison, the Marianas Trench is 10,994 meters deep and hosts some really strange life.
5.4 I don't know the chemistry but based on the Marianas Trench article in 5.3,
At the bottom of Challenger Deep, calcium carbonate shells are not an option because the intense pressure -- over 1,000 times sea-level -- dissolves the mineral.
coral reefs as we knew them will disappear. Extending this dissolving behavior to other minerals, it may not be possible to avoid significant changes to the topographic features of Earth.
5.5 Depending on how the aliens remove the water and where they suck it up from, there may be significant erosion as the water drains away. Erosion characteristics of megafloods will show up everywhere. To compound the problem, all the plants that used to prevent massive erosion are dead or simply disintegrated by water pressure.
(I'd like to thank Wolfram Alpha for making some parts of the calculations far easier.)