How to 'pick' transactions to include when attempting to add a new block?
Basically, you can include whichever transactions you'd like.

Currently most miners simply include all valid transactions they know of. It is cheap to do so.

There is currently a limit of 1 MB per block, so as transaction volume reaches these levels, miners will typically want to include transactions that have the best ratio of fee / data size.

As a more mature transaction fee framework evolves, miners will choose transactions to optimize for it.

Playing with transactions included is not needed to extend your work scope, as you can extend it indefinitely by changing the extraNonce.