Frequent, sudden and unpredictable BSODs

[DeadEnd]

After a prolonged period powered down, my computer boots up without fail and runs fine a few minutes, usually longer if not running a graphics-intensive game. But then, without warning, it freezes utterly, the picture disappears and the sound becomes loud screeching. Only a power reset helps. Then it will not boot up immediately (no beep from BIOS, caps lock light does not toggle), and even when it does, Windows might not load. Earlier, it would do this only for a couple of times and then stay up indefinitely but the situation is deteriorating, now it is hardly usable.

Plenty of different BSODs, appearing after a freeze about half the time:

MEMORY_MANAGEMENT
PFN_LIST_CORRUPT
SYSTEM_SERVICE_EXCEPTION
BAD_POOL_HEADER
DRIVER_IRQL_NOT_LESS_OR_EQUAL (atikmdag.sys)
MEMORY_EXCEPTION
POINTER_REFERENCE
PAGE_FAULT_IN_NONPAGED_AREA
Stop error
CMOS checksum error
faulty driver on the kernel stack

Memtest86+ reports no problems. Temperatures seem fine, and occasionally the computer runs for hours with no problems, even with a fullscreen DirectX game. A repair shop was unable to find fault, although their dust-clearing may have helped. All Windows updates installed. No hardware changes whatsoever. I have not been able to narrow down the problem.

This is really aggravating and I'm really lost with this one, please advice!

 

[DeadEnd]

Attached DM Log Collector results.

 

[DeadEnd]

New symptom: "BOOTMGR image is corrupt. The system cannot boot."

This is a strange problem... if it was just software (drivers), why would the length of the interval between bootup attempts matter? But if it's hardware, which component?