Yes, you can play PSP games without a memory stick. However, if you're playing games from PSP UMD's (little PSP games in cases). If you want to play games saved in your PSP's hard drive, then no. So, pretty much by UMD, you can play games without a memory stick in your PSP, but you'll be unable to save.
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You can copy games from one PSP memory stick to computer and then copy to another PSP's memory stick. Make sure you put the game on the folder 'GAME' on the memory stick.
yes by buying it on the PlayStation network and putting it on the memory stick
You will need a memory stick for ALL things you want saved on the PSP.
You just need a Memory Stick inserted and if the game you play offers game saving features (most if not all do), it will detect the Memory Stick and write to it directly.
When you download a game from the PlayStation Network, they are stored on the PSP's memory stick.