Read GnuPG encrypted emails transperently on iOS
Do you need to use Copy &amp; Paste with PGP/GPG mail?

No, that is not required.  For iPGMail:


Click the encrypted attachment to open it - you see the encrypted
text. 
Click on the transfer button (at top right). 
Open in iPGMail.
Enter your pgp password and see the unencrypted message.


Do any of the tools integrate like GPGTools on OS X?

This is impossible given Apple's sandboxing requirements.  As an aside this may well be a good thing as it avoids the annual scramble by GPG developers when Apple updates OS X and its Mail app.