I think the lack of a discrete graphics card may be your issue. DOTA 2 is based off the Source engine which requires a discrete graphics card, or a fairly powerful integrated card to run.
CS is a lot older and based off Half-Life (which is really old) and probably has significantly lower requirements.
DOTA 1 is a Warcraft 3 mod, and has relatively low system requirements. That said, I had a proper graphics card running it back in the day. Integrated graphics just aren't supported as well.
System requirements for Dota 2:
Operating System: Windows XP, Vista or Windows 7
Processor: Intel Pentium 4 @ 3.0 GHz or AMD Athlon equivalent
RAM: 1 GB
Hard Disk: 4 GB of free space
Video Card: nVidia GeForce 7600 GS or ATI Radeon X1600 series
Sound Card: DirectX 9 Compatible
DirectX: Version 9.0c
Broadband Internet Connection
Requirements for HL1 (the 3D engine CS uses)
Minimum CPU Type: Pentium
Minimum CPU Speed: 133 MHz
Minimum RAM Required: 24 MB
Minimum Hard Disk Space: 400 MB
Graphics Type: SVGA
Graphics Resolution: Multiple Resolutions
Color Depth: High Color
Requirements for Warcraft III (which dota is a mod of)
400 MHz Pentium II or equivalent
128 MB of RAM
8 MB 3D video card (TNT, i810, Voodoo 3, Rage 128 equivalent or better) with DirectX® 8.1 support
700 MB HD space
4X CD-ROM drive
As you can see, they are vastly different.
My recommendation is really to get a graphics card.