Why couldn't they just invade on a particularly stormy night or in particularly foggy weather when the RAF would be unable to find targets, nullifying their air advantage?

Considering that most of the craft designated for a German cross-Channel invasion were river barges, stormy weather would have been disastrous. Foggy weather made navigation extremely difficult--which would have been a hazard to the invading force.

was it so weak they couldn't take and hold a particular shipping channel for as little as a day to get a significant landing force into Britain in the same way that D-Day did for France?

Yes, the German navy was that inferior to the British. The gap between the two was massive. The Germans had four battleships, the British had over a dozen. The Germans had zero aircraft carriers. It wasn't close. (See the post from /u/Bigglesworth_ below)

why was crossing such a small sea seemingly an insurmountable obstacle when conquering thousands of kilometers of Russia against millions of men and fighting a war in North Africa were seen as achievable goals by comparison?

It's not just getting the troops across 22 miles of water, it's supplying them once they're there. That was the biggest feat of D-Day. You don't have to hold the Channel for a day or an evening to get your troops across, you have to continue to hold it for weeks afterwards--and you have to be able transport thousands of tons of supplies continually during that period. Without supply, an infantryman rapidly becomes a starving man with a poorly shaped club and a tank crewman becomes a thirsty man with a poorly functioning if remarkably durable house.

In the face of the RAF's existence and the dominance of the Royal Navy, any proposed German cross-Channel invasion faced obstacles that were nearly insurmountable.

Because its very hard to "sneak" 10'000s of troops across the channel without being spotted. Also the Germans troops would be carried across mainly in converted inland river barges that would have to be towed by tugs. Hardly suitable for traveling during a storm and crossing during fog would increase the chance of a collision, friendly fire or even missing the beaches entirely.

Also the allied navy didn't just hold a shipping channel for a single day, they'd brought a fleet of 6,939 ships over 1200 of them warships and established total control of both the sea and air around the landing zone until the end of the war. They also build artificial harbours and a pipeline to keep them supplied. The German's could never have matched such a effort, either in scope or in technology.

Its was much easier to advance across Russia because the German army had been built for ground warfare. To advance you simply had to keep driving without having to stop, move to another vehicle and then carry on, then get out again with even less stuff and be told to keep on advancing.

As others say, one of the key issues with modern warfare, from say the 1800s onwards, is the ability to supply and victual your troops.

If all your troops have is swords then food is the only real issue, and 20,000 men can forage for a while. Modern warfare however involves hundreds of thousands of men and a requirement for massive amounts of ammunition and fuel.

The germans would have had no train network to supply them as they moved inland, so everythign would have had to be brought forward by truck and horse. For efficiency they would have had to largely rely on their own supplies of both, including fuel for the trucks (and tanks). ALL this would have to be brought across by boat, along with ammunition and with the troops. And then constantly resupplied.

One of the miracles of DDay wasnt the (fairly) easy assault, but the mulberry docks and eventually PLUTO, a pipeline to bring in fuel, which enabled the massive amounts of supplies to be brought in.

In Germanys case this would have required vast amounts of highly vulnerable shipping, between the RAF and RN they would have been decimated.

So an early Sealion would have been a difficult assault followed by a nightmare supply scenario.

The people in command knew full well that their main issue was the long term supply of forces, not the putting of them on Englands shores.

isn't the channel as little as 22 miles wide?

At its narrowest yes, but the German plan called for the invasion to be launched across a much wider front than that.

Some of the proposed invasion routes would have been very much longer. Some routes were even from Rotterdam to the east coast of Kent. A distance of around 125 miles.

Le Havre to the Sussex coast is about 90 miles.

In addition to this, factor in the tidal flows in the channel, which are typically about 2.5 mph.

Now, even if we are quite generous and allow a speed of 5 mph for the mostly towed river barge of an invasion fleet, and allow for roughly 12 hours of daylight in mid September, you can see that there is really quite a big problem here.

Yes indeed, because its going to take more hours of darkness, and/or more more hours of daylight than there are available, in which to make your crossing.

So what to do?

• Leave at dusk, risk a night crossing, and hope to arrive in broad daylight, probably about tea time, after being attacked for hours by a now fully alerted enemy?

• Leave at dawn, spend the whole day being attacked by a fully alerted enemy, and try and make landfall at night?

• Maybe invade Russia instead?

The reason the allies were able to pull it off in 1944 was because they had absolute naval and aerial supremacy that prevented the enemy either knowing the invasion fleet was at sea, and doing very much about it even if they had known.