Solid Hardwood, Engineered and Laminate Flooring - undercutting door jamb

I am interesting in putting laminate flooring down and I have questions and concerns.

What other more common (cheaper) tool can be use to undercut the door jambs. I have a dremmel type tool and a hole cutting saw, router, circular saw. I don't need another specially saw for a few door jambs. I know about the flush hand saw but would have to buy one.

Where would you start laying the floor in an area that has two sections divided by stairs going down in the middle of the two sections. I have read that you need to worry about the stairs-you wouldn't want a small piece there but it looks like it would be hard to make sure that the ends of both rooms would be kept fairly square. Keeping it all fairly square boggles my mind anyway.

Our main support beams under the floor go the width of the house and I think the underlayment went the other way. Which way should my planks go-with the support beams or across the beams or does it matter. For some reason I have read that you should also take into consideration how the sun comes into the room-they should lay long end toward the light but I like the look of the light across the width of the boards-shows better grain.

I like to find the answers to possible problems before I begin a project.

Stanley makes a flat saw similar to Japanese saws that will do the job nicely. Cheap, available at most home stores.