Am I getting a better deal?

You are a ways off in your knowledge of the CPU.

i3, i5, and i7 designate the architecture of the technology and features of the CPU's thus named.
i7 is the top of the line intel, and there are levels of speed, features AND core counts within the i7 lineup. The 7 does NOT refer to the number of compute cores within the CPU.
i7 CPUs have what is known as 'Hyperthreading". This technology allows each core to run two 'threads' of information at once, thus simulating a second core. In "Device Manager" a quad core (that is 4 cores, not five) would appear to be 8 cores or processors when hyperthreading is enabled.
i7 cpu's can be found with 4 or 6 actual cores with a large range of features other then hyperthreading

i5 cpu's are the next step down from the i7. They do NOT have hyperthreading. In the Intel lineup they are probably the best bang-for-your-dollar that you can get. They are available in dual (2) core and quad (4) core versions, at various speeds (GHz) and prices.
In the i7 and i5 lineup, a K designation at the end of the CPU number (such as i5 2570K) denotes a cpu specifically unlocked for overclocking.

The i3 lineup has the least features, but are still very capable and powerful processors. They DO have hyperthreading. They are available in dual core only as far as I can tell.

Your Celeron is economy version of the various Intel processors. The speed is fine, but that doesn't tell the whole story about how fast or well it will run. Very briefly, and without detail, the instruction sets for calculating information in the Celeron are much more limited then the i3,5,7 cpus, as well as the memory cache on the processor die itself. The cache size affects performance; larger is better.
I am sure you will find your new Gateway considerably faster then the old emachines, even though the GHz speed designation is slightly lower. The integrated graphic probably won't work well for gaming or video streaming, but your old card will help, though you might want to upgrade the video card as a 430 probably won't be able to meet the power of the new cpu and may be the bottle neck in gaming or streaming. An Nvidia GTX 650 or GTX660 would be a good match.