I can't speak for SEP, but SEE can be pushed from just about any 3rd party software distribution product. The documentation even covers how to initegrate with Active Directory and use GPO's to push out the client installs and upgrades to your environment. Below are the answers to your questions.
Q1. I do not know what distribution software is being used but am curious as to how many devices can have their new software installed concurrently?
A1. The only limitation would be with the distribution software itself. There is no limit with in SEE.
Q2. My understanding is that both the SEE and SEP can be pushed in the background so there would be no down time for users - true or false?
A2. This is true for SEE and SEP.
Q3. Many users are not wired to the network - what are the possible challenges when dealing with devices connected to the network wirelessly?
A3. This will not make any difference as long as the clients is connected and the distribution software is configured to work with lower bandwidth devices.
Q4. I was told by a person claiming to have extensive knowledge about software pushes that the device does not have to be turned on to succesfully install the software as long as the network card is active and that the LANDesk techs have the ability to activate the network card remotely - true or false?
A4. I really don't think this will work. Unless the machine is configured to "wake up" on network activity.
Q5. Some of the laptops that have already been upgraded to SEE and had the old software removed are showing up on both the LANDesk reports as the old encryption and on the SEE console as SEE. I'm sure the team involved will figure out why this is happening but thought someone in this group might have run into this before and know the problem/solution. Maybe a fragment of the old encryption software left in the registry?
A5. Without understanding more of the environment, it will impossible to diagnose this. However, your thoughts that is could be something left over in the registry is possible.
Hope this helps.