Does the head coach "style" really influence the playbook? The last game, I had 15 rushing attempts for 40 yards against more than 60 pass attempts for 444 yards! That looks more like a Kurt Wagner team than a "run focused offense"!

The style determines the plays that are in the playbook - a run style offense will have more running plays, west coast will have more short passing plays, etc. The style also determines the default attribute weights to make the composite rating (i.e. a run offense coach doesn't value the QB as much as the RB). You still control what plays are called and also can always override the coach's opinions about the attribute weights, so what it really boils down to is the plays listed in your playbook.

I hired a HC who runs a 3-4 instead of a 4-3. When do the 3-4 defensive alignments show up in the available defensive plays?

I hired a HC who runs a 3-4 instead of a 4-3. When do the 3-4 defensive alignments show up in the available defensive plays?

As soon as he's listed on your staff the playbook should reflect his. You may still see some 3-4 plays but the majority of the base defense plays should be 4-3.

If my DC likes to run the 3-4 but my HC likes the Blitz heavy 4-3 who ranks on the playbook? whats play will be run is it better to have a 4-3 DC?

The HC defines your playbook - the other coaches have those values so you know who you might want to go after for a new Head Coach.