You can download the app or use your PC, personally I am not a fan of using the phone app for that stuff, a lot easier to lose your phone then your PC.
Yes, you purchase on coinbase ---> send to blockchain ---> send to source. Some people bounce it to multiple wallets first for extra protection, or some people totally "wash" their coins by using applications that split up your payment into a bunch of little ones and then send it for a small fee.
But the whole thing that protects us from being scammed when we purchase BTC -- the public ledger -- is what is always traceable back to the wallet. However the address that you send from is what it can be traced back to, not so much you. I am over simplifying here for explanation purposes.
The reason it's a bad idea isn't so much LE but the fact that Coinbase can shut down your account if it feels you are sending to less than reputable wallets. Then you would have to go through the trouble of making a new account, linking your card/bank account, and verifying your ID all over again.
If you are going the route we suggested (using your bank account with the wait versus using a debit for instant + fee) then no it generally always takes 5 business days or so. That's why we recommend sending the money to cash versus buying the coins. Once you have the cash in your account you can buy the coins as needed and not worry about price manipulation.