Since the succession law divided the inheritance among all sons, yes the empire gets divided. I could certainly see them trying to fight each other to get each other's territory.
Byzantium isn't going to take over Egypt anytime soon, after the initial Arab invasions they never went much further than Antioch, let alone Egypt. And the ERE had some troubles with the Seljuks and Normans at that point so it's very doubtful they can do anything.
Why don't we invade North Italy? It's rich and theoretically not impossible we succeed. Why not invade the ERE? It's rich and up until this point has proven itself to be weak.
Really this kind of argument can apply itself to a lot of places other than Egypt and they're easier to invade.
The US was not ready to take a million casualties to invade Japan, which is why they didn't IOTL and preferred sending nukes, asking the Soviets to invade Manchuria and watching Japan starve until surrender.
Anyways the Germans might have less troops at the beginning but in the unprobeable case...
It makes it significantly harder than if you were invading your neighbour.
Egypt is on the other side of the Mediterranean, I don't see why the Normans would plan to go navally invade it just because. The Normans didn't seem to have plans to do invade Egypt, for the matter there are several...
If the Soviets were to try to get something "substantial" they would risk nuclear apocalypse with the US. Most of West Germany is more than enough to get a favorable peace deal, which is all you can hope for in WW3.
It's not exactly the same thing as Lebensraum, Lebensraum wanted to eliminate all persons living in the East and replace them with Germans; the Pan-[ethnic group] considers that all these people are very similar and that they should live in one nation, not that they should be replaced. Those who...
Didn't fail because of that but fighting hundreds of miles away from your bases has some issues.
And you will notice the Fifth Crusade has bases nearby unlike the Normans here.
Why is going on land arduous? Striking at Egypt from Sicily is not very likely to be the Normans' first objective.
It would simply be suicide for France to go against all of the German Confederation. The British aren't going to support him since they don't want a war on the continent and even if they did, they likely wouldn't send much help to Denmark or France.
It's not like he can do much, he can't...
OTL they're guys with religious fervor who are doing something in the name of God and with his benediction, the Crusades are not something guys would do normally, it's an extremely risky thing to do.
Most Crusades didn't even go to the Levant navally but passed trough Anatolia, the only case...
Why would they go on an adventure on the other side of the Mediterranean? When they have several problems with the HRE even at home?
The Crusades went there because of religious fervor, not because it's practical or a smart idea.
Edit: Actually most Crusades passed trough Anatolia before going...