if this was my car i would run the blue wire(remote turn on)to the harness of the radio. looks so much cleaner and well done. i would not put the amp back on the box. doing that could/will hurt your amp. when it hits the wood flexes making the amp do the same. thats not good for the boards in the amp.
The wood does not flex, mdf wood absorbs vibrations which makes it a good candidate for sub boxes, mounting to the the box is fine I usually try to give an amp about 1 to 2 inches of minimum of clearance to allow it some airflow for cooling purposes, same for when I screw it to the box, I use 1/2" thick plastic spacers to give to some airflow room. Now as for the wiring, make sure that you disconnect the negative cable before you start removing wiring, that prevents shock or fuse pops when you are wiring making the job much safer. Next remove the connetions from the current amp and put them on your new amp 1 at a time to avoid mix ups putting them on the proper terminals, red being your +, grey being the -, Crimp on & solder some proper ring terminals to replace that hack job they did on the wiring there & attach it to the amp. I solder all my connections and I will give you a breakdown of how thats done after this. The small blue wire is the remote wire which turns on the amp, basically its just using a small amount of power to activate a larger amount of power and that should be attached to the blue wire with a white stripe on the back of you headunit, its usually marked remote or power antenna. If you want to run that to an accessory activation then please use a proper add a fuse circuit like the one bussman makes like the 1 here in this link
http://www.amazon.com/Bussmann-BP-HHH-ATM-Add-A-Fuse/dp/B000GKEXK2 they are the safest means of adding a circuit to your current fuse panel and they don't do any damage to the fuse panel like 90% of the current fuse taps like the 1 you have in the pic that you posted there. Just don't exceed 10 amps for either of the fuses. Lastly reconnect your negative terminal and enjoy the new amp. Now onto soldering.
Check out this link for soldering properly its a skill that will serve you well;
http://www.popularmechanics.com/blogs/automotive_news/4213013.html
If you are wondering about my qualifications I am a certified Journeyman heavy equipment tech with 15 years of automotive experience and well over 5 years of installing and fixing sound systems that ppl have installed in cars. There is a difference between doing it right and doing it to just make it work. I have seen some pretty severe electrical damage to vehicles from bad wiring jobs, take my advice and do it the right way, its not a lot of extra time or money to do it right the first time around and it gives you the peace of mind that its going to last.