1 up to you. Be careful with spending untill you know you will go further in the hobby. But if you take that advice. You'll be in a very small group of hams. Most of us have bad buys.
2 don't. Not at this stage. Building an amplification circuit is way more complex then it looks.seperation and filtering of feed back in the system is a complex task not advised for a beginner. Sort of why projects don't include it. It drastically limits the people able to take on the project. Honestly a 2nd external amplifier project. Or purchased amp. Would be a better goal with way less risk of harm to others.
3 yes but other things then the house may be better. I use a 160m EFHW on my canal boat. With an adjustable unun 64/49/16/9:1 this and a mat auto tuner allow me to use all hf frequencies with low swr. I simply run a 10m pole off the side of the boat. And run the wire into the trees.
Remember anything like this works. But it is still going to be a compromis. But that is true of most multi band antennas. They will be better at some bands then others.
I'll also warn you. What you describe is refered to as a random wire. But is not random. To work with a 9;1 unun it actually has to not match any frequency used. If it is close to half or quater wave on a frequency. It will need a much larger unun. Hence mine going to 64;1
If you look online you will find recommended lengths. And a list of bands they will work on.
But for portable work. I found the mixed unun ability much more flexible.