That's the great thing about the A1181 MacBooks -- because a fair amount of the parts are consistent between the versions/model years, you can easily source parts from other types of A1181 to replace or repair the MacBook you have on hand. On the other hand, this can lead to confusion because the specs of a given MacBook are printed on the underside of the topcase under the battery. So if someone had a 2.0 Ghz MacBook 1,1 (32-bit Core Duo/GMA 950) and used a good topcase from a 2.0 Ghz MacBook (5,1) (Core 2 Duo/GeForce 9400m), the person who bought that MacBook could in turn think they have a machine that's very different from what they really have.
To answer your question, re: the display -- I've heard a lot of people saying that a bad/failed inverter board is the cause of screens dying on A1181s, but in my experience it's almost always because the CCFL backlight in the LCD has died. Given the age of these machines, that's hardly surprising.
The only way to really know for sure is to take apart the machine and run it with a known good inverter board (which can be ~$15-20 USD on eBay). If it doesn't work, then you'll have to replace the entire LCD itself (which can be ~$20-80 USD on eBay, depending on the condition and seller). I've often found that many sellers sell entire LCD assemblies (which is the LCD, inverter board, display bezel, LCD case and WiFi antennas) for less than the cost of invidivual LCD panels. So you may want to go that route to save money and time.
As for the power button, that's tricky. It's possible that the keyboard connector could have been damaged when the topcase was replaced, leading to the power button not working. However, in my own experience, it's more likely that the solder joints for the power button may have been cracked due to rough usage (i.e. from previous owners pushing hard on the button to reset the SMC, or reset the Mac). You'll likely have to replace the entire topcase then; replacements can range from about ~$20-60 USD, depending on the condition and seller. If you don't mind the extra time in waiting, you could also try replacing the keyboard ribbon cable first. Those cost about ~$6 USD.