Strip flooring can be installed in any direction in any room, however the convention is to run parallel with the longest wall in most situations. Rarely is this type of flooring run perpendicular to walls in a hallway. Not only does it tend to make a hallway look narrower than it actually is, but can cause a lot of waste of material when installing. Don't be afraid of changing directions from the room to the hallway if your using a good wood product that can be cut cleanly on a table or chop saw. You simply need to make a very straight and slightly back angled edge (maybe 2 degrees) so the adjoining 90 degree slats are tight with no open gaps. This is extremely difficult to do with cheap laminate, but not so much with a good wood product.
There are transition pieces available for both styles, but they usually involve a change of floor level like a threshold and are not a good idea to use on open flooring, as they can create a tripping hazard, however they are fine to use in doorways etc. If you really want to have different directions between spaces, consider using a darker color "frame" around the room. This can make a very attractive transition to a large hallway or separate use area in an open design.
Let me add that I am NOT a fan of laminate flooring. Laminate flooring comes in a wide variety of quality levels, from extremely poor to fairly good, but share a common trait. They are always a "picture" of wood on some pressed paper or synthetic backing. They can be miserable to work with, easily damaged and almost impossible to repair after installation. Consider an engineered wood "click lock" instead. Pricing is very close and the engineered products usually are much more stable due to having a solid plywood backing, and somewhat refinish-able with a true hardwood veneer. They also handle the occasional water spill much better than most laminates that usually swell, bubble and blister at the site of water!
Another serious consideration is what type of flooring to use in a large high traffic area vs a smaller room. Laminates are almost always floating and will invariably sound hollow and creak as the temperature changes. For instances, walking across a cold laminate floor first thing in the morning can be like entering a haunted house at a fun park! Seriously investigate the advantages of a better quality engineered product.
The 1/4" plywood is likely luan. It's used to level the floor, often for a vinyl floor. For a hardwood floor, the main reason to have luan is shore up the floor and reduce any pitting or other slope issues. Whether you can remove the luan depends on the quality of your subfloor. If it has any imperfections, I would keep (or replace) the luan.
Best Answer
There is no way I would use a cork underlayment to make the floors equal. Cork has too many expanding properties. I don't even think you could get the cork to "sit" under a thin laminate. Also cork will feel softer than most other forms of underlayment. To the point where your floors will feel different.
I would use plywood or what is under higher floor (probably plywood).