PC Freezing at Startup Screen (or not), Occasional Display Crash, Motherboard Beep, Or Working Fine?

Very unusual problem here I can't seem to figure out. Any help would be appreciated. Here's what I'm running (cut and paste straight from system information):

OS Name Microsoft Windows 7 Ultimate
Version 6.1.7601 Service Pack 1 Build 7601
Other OS Description Not Available
OS Manufacturer Microsoft Corporation
System Name COMPUTER-PC
System Manufacturer System manufacturer
System Model System Product Name
System Type x64-based PC
Processor Intel(R) Core(TM) i7-3820 CPU @ 3.60GHz, 3601 Mhz, 4 Core(s), 8 Logical Processor(s)
BIOS Version/Date American Megatrends Inc. 1103, 4/9/2012
SMBIOS Version 2.7
Windows Directory C:\Windows
System Directory C:\Windows\system32
Boot Device \Device\HarddiskVolume1
Locale United States
Hardware Abstraction Layer Version = "6.1.7601.17514"
User Name Computer-PC\Computer
Time Zone Eastern Daylight Time
Installed Physical Memory (RAM) 32.0 GB
Total Physical Memory 32.0 GB
Available Physical Memory 28.2 GB
Total Virtual Memory 64.0 GB
Available Virtual Memory 60.1 GB
Page File Space 32.0 GB

Now here's the problem. It started with an update to my Geforce GTX670 about eleven months ago. I'll explain this step-by step.

1. I'm running my PC into a Samsung HD TV.
2. At first I was doing this with a DVI to HDMI cable.
3. After the one Geforce update, upon starting up the motherboard would give a series of beeps in succession and nothing would display. Three of them, one long, two short.
4. Nothing would display on the screen other than the Samsung TV having a "no signal detected" message. Going to a regular PC screen like I had originally did not solve the problem.
5. Strangely, when I swapped the DVI to HDMI for a DVI to VGA cable, this completely solved the problem for about seven months.
6. I noticed as the PC started up that the TV screen would display three different resolutions, as though the Geforce was cycling through them, and it would stop on my current resolution, which is at 1366x768. Not sure if this is important, but it's strange to me.
7. Eventually the problems started again, which takes one of three forms, none of which seem to occur for any particular reason. a. The computer will sometimes just freeze and then crash with any sounds that were playing (music, a game, etc.) buzzing. b. The computer will freeze at start up, showing the Windows loading screen and even playing the "welcome" music as though it was loading up to the main desktop, but staying on the start up screen. c. While loading the motherboard will beep as mentioned above, displaying nothing. d. One time it crashed, showing the blue BIOS screen, and then restarting as if nothing had happened.
8. Otherwise it will operate normally. The following things have worked, but with no real consistency. Sometimes for several days, or even weeks. a. Playing with the angle of the VGA cable. b. Taking out the cable and putting it back in. c. Starting the computer several times.
9. I was beginning to think it had to do with the VGA cable itself and how it hangs from the computer (one screw is missing to hold it in), but then it worked fine for several weeks UNTIL the latest GTX670 update about five days ago.

So, what do you guys think? Any suggestions or additional information you require? I'm using it right now after about thirty minutes of messing with it, and it's suddenly fine again. The strange thing is it's not consistent, it will probably work fine for a full week now and then do it again. Thanks for any help!