power: you will need a 12v regulator-- cars voltages vary from 5-18 volts at times-- electronics don't like that much fluctuation.
i also prefer to run a new power line from the battery instead of tying into factory wire for any of my add-ons.
this has some advantages over the factory wiring-- you know exactly what is getting power from that line(in my car, some fuses are for more then they are labeled for, so the fuse rating is less then expected when i try to replace 1 factory device..) and if you do blow a fuse, you will be able to trouble shoot a little faster because you know where the wire goes, and what it powers.. and because it was added on, it should not ever affect any of the OE vehicle stuff, so you don't have to worry about messing up your car as much. you can also hide the wire pretty good, and run it anywhere you need it in the car!
for the location, i doubt that under the dash will have enough room, but if it does, it will be a very tight fit, and you will need to consider ventilation.
under the seat is a good spot, assuming that it will fit (trouble points can be the seat rails, and seat-bottom-to-floor clearance) the biggest problem here is that you need to remove the seat for servicing, and that there is a possibility that stuff might get spilled on it..
depending on the car, there are sometimes large areas behind factory panels in the trunk that can work well, or sometimes, you can mount it below the rear parcel shelf with a small amount of difficulty..
you should definitely be able to find a spot in the car that the laptop fits, and hides good in.
oh, and for wiring, check under your door sill plates-- those plastic covers at the bottom of your doors, next to the seats-- every family car i have worked on has more then enough room in there for almost any wire you need to run--and be completely hidden
for the overall setup, i am not sure how easy to use that android app is(can you operate it at 70mph with minimal distraction? not all apps take this into consideration), and for me, overall phone connectivity is sketchy, so i wouldn't want to rely on my phone connecing to my carpc.. another thing you might want to consider would be a display like this to see what is playing:
the typhoon series is at your budget limit(i don't play the penny-less game ), but last i looked at these, they are supposed to have the easiest software to work with.. but there are cheaper options that can do the same thing..(just look around their site)
and if you decided to scrap the phone-control thing all together, you could use something like the griffin powermate to control the songs..