Why does my wallet need to download the whole transaction history? I started a wallet, and it is downloading and synchronizing lots of transactions. 
This takes time and network bandwidth. 

Why is this necessary? Is this some form of accountability?

With Paypal, for example, this isn't done. 
And as you grow, I guess this becomes a huge -- will the amount of transaction data become too large? It is a bit scary to have my computer run off and synchronize, without it informing me  what kind of information it is getting and whey this is necessary.

Can't the whole transaction history be shown or available on a website, rather than loaded on my computer?
The short answer to your question is "no". The reason is actually pretty simple: Without the entire block chain, there is no way to know that funds that are claimed to have been sent to you weren't actually sent to someone else. There is no way to prove the absence of a conflicting transaction, other than to check every valid transaction.

With a bank, the bank clears your checks for you. Here, you clear your own checks. Without knowing every check an account has received or sent out, how can you know the account has sufficient balance to cover your check?

Of course, you don't have to run the Bitcoin client. Your alternative is trust someone else with your coins.