Can I use the same time machine backup for two macs? Is it possible to use the same time machine backup for two different machines? Is it possible to take the backup of my laptop and push it to my desktop?  If I make changes to one machine do the changes go to the other machine?

I know I can restart and hold CMD + R and select a time machine.  So far though the time machine backup is not recognized.  I ended up with two sparse files.  So all the programs and stuff on one mac are not installed on the second mac.
In a nutshell: No, you can't.

A backup is not a synchronisation solution so there is always a 1:1 relationship between your main drive(s) and your backup. You can use Time Machine to backup multiple computers to the same network drive, but the backups will always end up as separate bundles.

To transfer applications/data from one Mac to another once, have a look at Migration Assistant (/Applications/Utilities/Migration\ Assistant.app). To share data with ongoing updates from several machines, consider defining remote access/folder sharing on a master drive or look into rsync.