I am having some trouble understanding multicore CPU speeds. An E4500 is advertised as being 2.2ghz and has a lower clock speed than a p4 3.0 but will out perform it. Is it because a E4500 is 2.2ghz times, meaning it has 2 cores at 2.2ghz each.  
 
Or in the case of a quad core like the q6600, is it 4 cores at 2.4ghz each for a total processing power of 9.6ghz. Is my thought process correct, because when I go into the bios it shows my e4500 @ 2.2ghz, but is it really a 4.4 ghz cpu?

you have 2 processors running at 2.2Ghz, on one chip. that one Value in the bios is the set speed for all of your processors.
 
your total speed actually depends on the apps you run, they must support multi threading, if they dont your basicly running 1 core at 2.2ghz, while the other cpu is running other apps.

The processor frequencies are for each core.  So, yes, a E4500 has 2 2.2GHz cores.
 
Also, there was an architecture change, which resulted in significant improvements in performance per clock cycle.
 
The analogy you might be looking for is this:
p4 3.0GHz = 1 3L bucket
E4500 2.2GHz = 2 2.2L buckets
 
Of course, if you only have one person (progarm/thread) using the buckets, you'd rather the bigger bucket[1].
 
[1] The E4500 bucket is probably as big or bigger, due to the fact that frequency does not equal performance.

2 programs running at the same time. 1 program uses 100% of 1 core, the other program has to wait (thus those little hourglasses) 2 cores, 1 is free to work on the other program. Also, say an old cpu runs at 3Ghz, but only has 20 threads, or workable connections to every program. The newer one works at only 2Ghz, but has 80 threads. Old = 3x20=60 . New 2x80=160 get it?

 
First of all, more MHz does not mean more performance if we are not comparing the same line of CPUs. There are other ways to increase the performance of a CPU rather than to just increase the clock speed. The Core 2 line of CPUs is very different internally than the Pentium 4s and can do more work with each MHz than the Pentium 4s can. A single Core 2 core at about 1.60 GHz will perform similarly to a single Pentium 4 core at 3.00 GHz, even though the Core 2's clock speed is a little over half as high as the Pentium 4's.
 
The E4500 is a dual-core CPU, so it can also derive some extra performance from that second core if the application can use more than one core. Otherwise, one core of the E4500 is being used and the other is idle. So in some situations, yes, a 2.2 GHz Core 2 Duo will be twice as powerful as a 2.2 GHz single-core Core 2 and a 2.2 GHz Core 2 Quad will be four times as powerful as the single-core Core 2. But in others, all three chips will have the same performance- it all depends.

Yep, MHz doesn't matter as much as the CPU's architecture. AMD taught that to Intel, and Intel learned (grudgingly) that lesson very well.
 
Image two house of equal size needs to be painted; the owners of house #1 hired 3 painters. The owners of House #2 hired 4 painters.
 
The painters of House #1 (the old Pentium 4 core) needs to paint faster (more MHz) to finish at the same time as it would take the 4 painters to paint House #2. The painters of House #2 can paint slower to finish at the same time as the painters of House #1 because they have an extra painter.