to answer your questions in order.
Yes. Weapon of the Gods will work extremely well with your rogue powers. It will even add the radiant keyword to them. This is especially useful when dealing with undead. Your weapon remains a light blade/crossbow/short bow so you can definitely continue to inflict sneak attack damage.
Yes. The status affects inflicted do not conflict with the damage type. In fact the damage type is added to the damage type on the power (for instance if the power deals necrotic, it would also deal radiant).
Yes. The full version (not hybrid) of sneak attack requires you to hit with a specified weapon, not that it be a rogue power, so cleric powers are great (provided your valid rogue weapon is the implement or the weapon for the power). To add to this, the hybrid version of sneak attack (labeled "Sneak Attack (Hybrid)") does require you to use it with rogue powers and would not work in this case.
So yes, your planned build works just fine with the main class as rogue and a multi-class to cleric taking a cleric PP works just fine.
Warlords have Opening Shove (from the Martial Power book), which does, in fact, push.
Warlords who use a two-handed spear or polearm can also take the Longhand Student feat, which adds push 1 to Viper Strike. Warlords who use a greatbow or longbow can take the Adamant Arrow Student feat to add push 1 with Paint the Bullseye.
If none of those meet your needs, there are two ways for characters of any class to gain another class' at-will power. Both kick in at (or can kick in at) level 11, but also require a fair amount of investment.
One: Be a Half-Elf
Half-elves have the racial ability Dilettante, which lets them select an at-will power from another class and use it once per encounter. There is a paragon-tier feat, Versatile Master, that lets them use their Dilettante power at-will.
Two: Paragon Multi-class
If a character of class A multi-classes into class B, and has taken the encounter, daily, & utility power swap feats by level 10, then at level 11 they can choose to paragon multi-class (PMC) in class B instead of choosing a paragon path. Among other things, this allows them to, at level 11, replace one of their two class A at-wills with an at-will from class B.
Best Answer
Yes,
Sneak of Shadows increases as your level does. Taking a multiclass feat gives you the full rights and privileges of whatever you qualify for in regards to the additional class.
In this case Sneak of Shadows grants you the ability to use Sneak Attack 1/encounter as a rogue of your level.