Do you think my motherboard is bunk?

Hi, I'm not a computer guy. My system is almost 3 years old, but the last year I have had two problems I am starting to suspect are from my motherboard:

1) My internet connection will randomly lose connectivity to the net. It also has these constant lows like the connection is dropping slowly. The lan light is green and blinking (I think its supposed to be solid?) but also the US/Upstream light will turn orange most of the time. I called my internet guy and he tested the modem and cables and said it was my network adapters problem (on mobo). Whats odd is that when I speed check it shows a nice connection, but its obvious when playing multiplayer games or watching streams that its lagging constantly. Besides, a few times a week it will just stop and I get disconnected. All the lights will suddenly blink together like its trying to reconnect and then they go blank for a few seconds. It does this repeatedly for an hour or so and then suddenly 'finds' the connection again and I have internet access.

2) Once or twice a month, always during some memory situation (such as multitasking with different programs and clicking unresponsive tabs) the comp will freeze and go to a blue screen that says there is a hardware problem and windows is performing a memory dump. It always starts right up fine afterwards though. The latest time it happened I took a pic with my phone:

http://i.imgur.com/ktAmsQ6.jpg

Please advise me if theres a way to diagnose/fix the problem or whatever steps I should take.

Thank you for your guidance.

I used Belarc Advisor for this info:

OS:
Windows 7 Professional (x64) Service Pack 1 (build 7601)
Install Language: English (United States)
System Locale: English (United States)
Installed: 4/1/2012 8:52:23 PM
Boot Mode: Legacy BIOS in UEFI (Secure Boot not supported)

SYSTEM MODEL:
Gigabyte Tecohnology Co., Ltd. X79-UD3
Enclosure Type: Desktop

PROCESSOR:
3.20 gigahertz Intel Core i7-3930K
384 kilobyte primary memory cache
1536 kilobyte secondary memory cache
12288 kilobyte tertiary memory cache
64-bit ready
Multi-core (6 total)
Hyper-threaded (12 total)

MAIN CIRCUIT BOARD:
Board: Gigabyte Tecohnology Co., Ltd. X79-UD3
Bus Clock: 100 megahertz
UEFI: American Megatrends Inc. F7 12/26/2011

DRIVES:
7621.28 Gigabytes Usable Hard Drive Capacity
855.20 Gigabytes Hard Drive Free Space
ATA Hitachi HDS72303 SCSI Disk Device (3000.59 GB) -- drive 0, s/n MK0331YHGVW0PA,SMART Status: Healthy
ATA WDC WD6400AAKS-2 SCSI Disk Device (640.13 GB) -- drive 1, s/n WD-WCASY5617084,SMART Status: Healthy
INTEL SS DSC2CW480A3 SCSI Disk Device (480.10 GB) -- drive 2, s/n CVCV202400FQ480DGN,SMART Status: Healthy
Samsung M3 Station 3 USB Device [Hard drive] (2000.40 GB) -- drive 4
Samsung STORY Station USB Device [Hard drive] (1500.30 GB) -- drive 3, s/n 801130168383

LOCAL DRIVE VOLUMES:
c: (NTFS on drive 2) * 480.00 GB 72.17 GB free
e: (NTFS on drive 0) 3000.46 GB 4 MB free
f: (NTFS on drive 1) 640.13 GB 117.93 GB free
k: (NTFS on drive 4) 2000.40 GB 260.79 GB free
l: (NTFS on drive 3) 1500.30 GB 404.30 GB free

* Operating System is installed on c:

MEMORY MODULES:
16338 Megabytes Usable Installed Memory

Slot 'Node0_Dimm0' has 4096 MB
Slot 'Node0_Dimm1' has 4096 MB
Slot 'Node0_Dimm2' has 4096 MB
Slot 'Node0_Dimm3' has 4096 MB

USERS:
local user accounts last logon
XXXX 12/3/2014 1:09:06 AM (admin)
local system accounts
X Administrator 3/6/2012 7:53:14 PM (admin)
ASPNET never
X Guest never
X Marks a disabled account

DISPLAY:
NVIDIA GeForce GTX 560 Ti [Display adapter]
LG TV [Monitor] (52.0"vis, February 2009)
LG TV [Monitor] (7.2"vis, January 2010)

COMMUNICATIONS:
↑ Intel(R) 82579V Gigabit Network Connection
primary Auto IP Address: 118.139.202.242 / 24
Gateway: 118.139.202.1
Dhcp Server: 172.18.1.11
Physical Address: 50:E5:49:E6:8D:B9
Connection Speed: 100 Mbps
Microsoft 6to4 Adapter
Microsoft ISATAP Adapter
Microsoft Teredo Tunneling Adapter
↓ TunnelBear Adapter V9
Status: Cable unplugged
Dhcp Server: none responded
Physical Address: 00:FF:16:16:28:84
Connection Speed: 10 Mbps

Networking Dns Servers: 8.8.8.8
180.182.54.1
168.126.63.1
210.181.1.24

 

OMG. That image is huge, sorry. I can't find an edit button.

What is a good hard drive testing software?