During the 48h cycle with the fork chain.
The test chain now lasts as long as the third phase of the amendment process, so approximately 3 weeks.
They are simply processing the same transactions that mainchain are producing using the new protocol looking for bugs?
There is no particular reason to replay on the test chain the transaction that happen at the same time on Mainnet; it is OK if the two chains diverge.
Is there a specific procedure for the tests? Specific steps to follow?
There is no specific procedure, participants (not only bakers) are free to use the test chain to perform the tests they consider useful for them.
Most tests of proposed protocols can be run either in sandbox mode or on test networks (such as the Delphinet test network) long before the test chain is forked.
The main difference is that the test chain starts from Mainnet so the conditions of the activation of the protocol on the test chain are very close to what the activation of the protocol on Mainnet will be if the protocol is voted in the promotion phase.
In particular, bakers can get a good estimation of the time their nodes will need to migrate to the new protocol. This is important for the bakers and endorsers of the first few blocks after the activation of the new protocol. Testing this in sandbox mode is doable but not easy.
Application developers might also prefer to test their smart contracts on the test chain because the addresses of all accounts on the test chain are the same as in Mainnet.