You have to remove the HDD cover from the laptop. The HDD is right under that. Carefully remove it and hook it up to a SATA port inside your desktop. Make sure you connect the power cable to the HDD as well. Power up the desktop. See what boots up. It should boot froom this HDD if it had no other HDD inside. Due to a radical change in configuration you will most likely need to re-activate Windows.
If all works well, you could make this a more permanent fixture by using a HDD tray adaptor (to make it fit in a 3.5" bay inside the desktop).
no, you can not check the connection without opening the laptop, as described above.
Of course, this applies in case you want to use this as a bootable HDD. If you just want to connect it to the desktop to get data off of it, and hence use it as an external HDD, you could use an enclosure (still after you have removed it from the laptop, though. No way around that). For this purpose I would recommend a Thermaltake blacX that is like an opened enclosure. Quick access to the drive and excellent cooling. not pet-friendly, though.