I am interested in finding something that compare different beams. Simply put I have a 9 foot span that I would like to get the smallest beam to span. Currently there is a 4X10 (2 - 2x10s nailed). I would like to replace it with a smaller beam but I am not sure what is comprable. I know there are several types (paralam, I beam, glue lam, etc.) But which will give me the thinnest beam? I wild like it to be max 6 inches think. Thanks for any info.

Go to your nearest building supplier, ideally one that also fabricates roof and floor trusses. They usually sell beams of the kind you mention. Talk to their engineers who can take a look at your project and advise you as to what product will possibly meet your desires. Most likely it will be one of the engineered beams you mentioned ie paralam, microlam, glulam etc. Give them a call!

You're gonna have to use steel. Laminated veneer lumber is not made in depths less than 7 1/4 inches, I don't believe, but a fairly light wide-flange steel beam will do it easily. If the beam is in the middle of the room, and the structure above is at least 2x8's, you can even make the 6-inch beam flush, entirely hidden beneath the ceiling.

You don't say whether it's a floor or a roof above, nor whether any loads from the structure above are also coming down on the beam, so it's impossible to give you even a rough idea of which beam will work.

Consult a competent aerchitect or structural engineer for guidance.

I also have a joist that needs replace. It's the main floor one on my first story. The house is an old one and the top of that joist has dry rotted. What's my best step?

Almost no lumberyard has an engineer on staff. As a registered engineer, I am offended by lumberyard salesmen and the like who have taken a seminar to learn how to run a computer program and then allowing themselves to be called "engineer".
There are many factors in designing a beam - shear, bending, deflection, bracing, support, etc. Without understanding and evaluating the factors involved, a substitute beam can not be specified. In short, call a structural engineer.

For "farmchick 117" - you might be able to simply sister the existing damaged joist. If the damage is from dry rot, moisture, insects or other similar, you should have the surrounding framing inspected as well. Rare when damage such as that is limited to just one member. You will want to strengthen or replace the framing and also cure what ever is causing the problem.