Please allow me to clarify my response.
1. You stated that you formatted the drive which you are attempting to use. That means the data on it is not important. That makes it not a data recovery issue.
2. You stated that you are trying to use a PATA drive in a SATA laptop. The drive is healthy. That makes it not an HDD repair issue.
I told you that under virtually no circumstances, short of replacing the PCB board on the PATA drive and thus converting it to SATA, will you be able to use that drive in your SATA laptop. That is not a cost-effective method, but for some drives it may actually be an option.
No add-on converters will fit as there is zero clearance. It's very inexpensive to purchase a SATA laptop drive and call the case closed, which was what I suggested.
Example of a data recovery issue "I formatted the hard drive with my first-born child's pictures".
Example of an HDD repair issue "I plugged in a laptop power supply into my external drive and it went up in smoke".
At this stage, I told you what you needed to know, and further questions such as Linux and related can be answered but they are not exactly related to data recovery and hdd repair, hence that would be an off-topic thread from my perspective. You would get much better support from a dedicated Linux on external drives forum. I personally use Linux daily as well as Windows.
It is certainly possible to run Linux from an external drive. It is not advisable to attempt to run Windows from an external drive connected by USB. eSATA drives connected to on-board SATA ports via a SATA-eSATA interface cable should work, in theory. I will be testing that theory.
So, here's a possibility for you to prove this point. Open up the laptop, connect a regular 12" SATA cable to the internal SATA port inside the laptop's drive bay. Connect the SATA to ATA adapter. Connect the ATA (40pin) to Laptop ATA (44pin) adapter. Guess what, you just made a PATA laptop drive compatible with a SATA laptop. You could also do that but go a step further and use a SATA to eSATA cable, connect it to an eSATA enclosure, and then reverse convert the SATA back to PATA and then step it down to a laptop PATA interface.
You would use
http://www.newegg.com/Product/Product.a ... 6812203012 and
http://www.newegg.com/Product/Product.a ... 6822998008 (this exact unit is shipped with Salvation Data products, if I recognize it correctly)
You would also need a Molex-terminated power supply. The one that comes with a Coolmax CD-350, which I use daily, would be acceptable.
http://www.newegg.com/Product/Product.a ... 6812104062It will look ugly, it requires a form of laptop disassembly, but it could be done in theory. The drive definitely won't inside the bay flush as a new SATA drive would. That would be reminiscent of hooking up an old 25-pin SCSI drive to a modern Ultra-320 controller on an 80-pin SCA backplane. It can be done, but it's pointless for all practical end-user purposes.
I use a similar combination chain in data recovery scenarios because our special equipment has built-in PATA ports and we use it to repair SATA drives.
I did not intend to sound rude and thank you for asking it. I am including an explanation to your question in my book in the SATA vs. PATA chapter.
If you happen to find yourself talking to a data recovery professional and catch a glimpse of what we charge for recovering data, you will agree that the $60 or so that it costs to purchase a new 2.5" SATA drive is immaterial as compared to what we'll charge for our services. The prices usually start in 4 digits.