Hey Reckoning. Let me answer your last question first. Microsoft makes a special tool to read minidump files called Debugging Tool. If you do a search you'll easily find it. However, in order for this tool to read your files you'll also need the Symbol packages. There are a number of them and they are large in size. It takes a
long time to download and install. Trust me.
If you decide to go ahead but need some advice on how to use it let me know.
Now for your minidump files. One is 0x0000001E: KMODE_EXCEPTION_NOT_HANDLED
The Windows kernel detected an illegal or unknown processor instruction. A Stop 0x1E condition can be caused by invalid memory and access violations similar to those that generate 0xA errors.
The other four are 0xA and these are caused by either hardware or faulty drivers attempting to access a higher IRQ address than it is designed for thus causing system crashes. Unfortunately all five files only cited a Windows OS driver and the thing about OS drivers is that they point to the fact that there is a problem without (most of the time) telling what the cause is.
The fact that there was an infection and in light of your symptoms and the fact that you can only run it stable in Safe Mode has me a) suspicious that you may still be infected and/or b) the OS has been compromised.
* I want you to do the following: Go here
http://www.eset.com/us/online-scanner/ download and allow the Active X. This is an excellent scanner and I personally have run NOD32 as my antivirus of choice for 6+ years. Let it run and get back to us with the results.