It depends on what you mean by "good for gaming".
The 3570k is cheaper than both of those by a pretty solid margin and pretty much everybody considers those to be excellent when gaming. I will use that as a baseline.
Going from 3570k to 3770k you get only a couple things, primarily:
1) 0.1 GHZ higher base clock speed.
2) 2 MBs of L3 Cache ( 6 --> 8 )
3) Hyper Threading
If you are only looking at gaming, you can pretty much write off #3. Almost no games support more than 4 cores anyway and almost no games support hyper threading anyway even if they did support more than 4 cores.
#1 is also kinda worthless because the 3570k can be OCd to whatever the 3770k can be OCd to. This would only be important to someone if they didn't want to bother with even a 0.1 OC.
#2 is the bigger thing, because the effect of this can't possibly be emulated on the 3570k. The 3770k will always have the advantage of 2 MBs and its impossible to add the other 2 to the 3570k in any way.
How much performance difference is that going to actually give you when you spend your extra $80ish to get the other 2 MBs of L3 cache? Some benchmarks are available here:
http://tinyurl.com/923d8x2
If you look through there, the 3770k usually has a low single digits FPS lead. In only one of those games does the 3770k's advantage matter even one iota. That is Starcraft 2. Everything above 60 FPS is worthless, because the human eye can't see faster than 60 FPS (delivered smoothly, anyway).
Starcraft 2 goes from 52.7 to 55.7 when the processor is changed from 3570k to 3770k, which is going to be noticeable to some low single digits percent of people. However, this same performance gain in the game could be had with just a small OC on the 3570k. Bringing it up to somewhere in the 3.8 - 4.0 range is very doable even on stock cooling and that should be enough to get the extra 3 FPS to close the gap.
The 3770k could be OCd the same way, but both processors would soon get into the zone where it doesn't matter what you do because you are at 60 FPS anyway and all the gains stop there.
Anyway, if you don't want to OC, then just be aware that the 3770k is not worth paying for most of the time, but in a very small minority of games if you are one of the small minority of people who can notice a 53 - 56 FPS move, then you might get some tiny benefit for the extra $100.
Personally, I wouldn't say that its worth it to pay 40% more for a 3770k over a baseline 3570k, because in the overall scheme of gaming things your 40% more expense buys you less than 1% more performance on average.
Luckily for us, the 3820 is also benchmarked on the same page, so I will compare the 3570k to the 3820 using the same information.
About half the time in those games the 3820 actually falls behind the 3570k. The other half the time its usually not farther ahead than the 3770k is anyway. Basically, if you don't think the 3770k is worth it you probably really won't find the 3820 to be worth it.
The biggest advantage that the 3820 has over the other two is that it has 40 PCIE lanes built in. If you are going to have tons of PCIE slots and have expansion cards in all of them, this kinda matters. If you are like a normal person then this doesn't matter at all.
The 3820 can easily SLI/Crossfire at x16/x16, which would use up 32 lanes, however, the performance of x16/x16 is sometimes even worse than the performance at x8/x8 and its almost never hugely better, so 40 lanes is really far too much for most people, even when they want to use multiple video cards.
If you seriously intend to use 3 or even 4 video cards, I would highly suggest leaning towards the 3820 out of the options listed, though. However, moreso I would suggest the 3930k over the 3820 if that is indeed what you intend. Even more than that, though, I would rather just talk you out of using more than 1 video card anyway.
As long as you don't absolutely need the extra lanes, the 3820 is worse in most ways. It is from a much older generation and uses almost double the wattage, makes a whole lot more heat, and everything that entails. It does have 10 MBs of L3 cache, though, which is kinda nice to have, but it really doesn't make up for just being architecturally behind the curve.
Anyway, looking at all these numbers I must say that neither the 3770k or 3820 appears to justify the added expense over the 3570k.
Asrock motherboard - Asrock has a pretty short track record as a stand alone company, but in that time they have done pretty well in terms of quality and especially in terms of giving more freebies for the same price as its longer track record brethren like Asus and Gigabyte. On that basis, I can't suggest avoiding this manufacturer out of hand.
Can I just suggest avoiding the extreme 6 in particular? Not really. It is kinda high on price. Most people do just fine with $130 boards and don't need to spend as much as $190. The extreme 6 has 8 + 4 voltage regulators which is something OCers like to see, but for a non-OCer it doesn't matter very much.
The extreme 6 also has a lot of video ports on the back of the PC which are nice for those people who: 1) don't want video cards, or 2) have some kind of video card problem and need the ports for testing purposes. #1 definitely doesn't apply here and the extra ports above and beyond those included in a $130 board doesn't justify the expense at all.
You probably get a couple more USB ports than the $130 board will give you, but using a lot of USB stuff at once isn't a really great idea anyway. A lot of people like it, but in the long term it does bad things to PSUs.
Anyway, I would say a $130 board will probably do just as well for you as the extreme 6 will and it would reduce the PC cost by another $60 in addition to the $80-$100 from above.
Corsair H100 - A big pain in the yin yang and usually provides no better performance than high end air coolers like the Hyper 212 Evo which are priced at like $50-$70 less.
CD Drive - the Asus DRW24-B1ST is best in class. If that is what you meant you intend to get, its a very good choice.
Case - HAF X is extremely good. I have never heard anyone complain that had it and everyone drools over it. I can't fault that choice.
PSU - HX 750 is a solid model, no complaints.
RAM - I would say that you should probably reconsider on this purchase. Two sticks are usually better than 4 for a few reasons that I don't really want to bother delving into right now, so if you wanted 16 GBs of RAM I would suggest you aim for 2x 8GBs rather than 4x 4GBs if you really want to have the whole 16 GBs.
However, nobody afaik has ever noticed the difference of having more than 8 GBs in gaming situations. You could get 2x 4GBs and be just fine if all you are doing is gaming. That would cut off another $40 or so from the PC expense.
Windows uses like 2 GBs of RAM itself, most of the time the other programs open when gaming can't get anywhere close to 2 GBs of RAM used and almost no game uses more than 2 GBs of RAM so that means most people can get by just fine with 6 GBs.
Considering its good to match sticks, its better to have 2x 4GBs than 1x 4 + 1x 2, so that is why 8 GBs is usually suggested over 6 GBs by most people when 6 GBs is usually more than good enough.
I personally have 8 and I don't think I have ever used more than 5.
Also, Corsair isn't my favorite RAM brand. They sell a lot of units, but they have higher failure (DOA) rates than other major brands. I would get Crucial instead, or maybe Kingston. Both of those two makers are tied for the lowest failure rates.
2x 4GB Crucial Ballistix 1600 mhz RAM is a great set of sticks and I would highly recommend them. I use it myself and they just slide in and work.
If you go with my suggestions from above, namely:
3570k over 3770k
$130 motherboard over $190
$30 air cooler over $100 water cooler
2x 4GBs Crucial RAM over 4x 4GBs Corsair, $40 vs ~ $80
Then you can take that savings of about $250 - $300 and get you a nice fat 256 GB SSD, which will definitely make a difference in the way you game and interact with the computer. A positively huge difference.
The same difference would be present if you had a 128 GB or lower SSD too, but 256 is way better than 128 and lower for one reason in particular. That being that its highly unlikely you will run out of space.
At the base the OS and the page file have to be on there which is 20 + RAM GBs (assume 8) and then you have to leave 10% of the drive free for most efficient operations, so that's another 13 gone (assuming 128 GBs). If you want to install some productivity apps like office and PDF readers that might be another up to 10 GBs gone.
By the time you get all that taken into account you are sitting at less than half of the 128 GBs available to actually install games. With many clocking in at 10 GBs or more each, that doesn't leave room for more than maybe 5 or 6 games.
Running the numbers with 256, though, you can get like 14 or 15 on there and not really have to worry about uninstalling the ones you don't play so often but still haven't decided to completely be done playing yet.
If you only ever have 2 or 3 games installed at once the difference between 128 and 256 is pretty non-existent, but if you like to have 5 + installed then its a huge difference because you aren't forced to uninstall the old to make way for the new. You uninstall only when you want to with the 256 and that is pretty nice.
Anyway, hope some of that helps you somehow.
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