The 7750 only draws power through the PCIe slot, so you don't have to worry about your power supply connection.
Yes, 300w is more than enough. Even though it's doubtful your power supply can even put out that much power, you should be just fine still - the 7750's power draw is 43w maximum. That's the IDLE of more power hungry cards. (http://www.techpowerup.com/reviews/ASUS/HD_7750/24.html)
Any 7750 will fit just fine - you've got a normal ATX mid tower there, not a slimline, and yes, you have a PCIe slot. I personally would choose this one for noise, as it's totally fanless:
http://www.newegg.com/Product/Product.aspx?Item=N82E16814161417
That being said, make sure that whichever one you get has the display out that you need to connect to your monitor - most of them have a displayport, an hdmi, and then either a dvi or vga.