It does. After the download is finished but before the actual installation begins, find the installer in the Applications folder and make a copy of it. If it's used to upgrade the boot partition, it'll be deleted afterwards.
No. The ill-designed app resides in /Applications and is coded to blow itself away during installaiion. You have to quit the installer and save a copy to another location. Then, you can eliminate wasting bandwidth, time, and electricity by copying it to other machines.