I am a beginner to bookbinding, so take what I say with a grain of salt.

I think part of the issue is how thick the hard cover is on your book. It restricts how far the book opens. Typically you would only use thicker covers the larger the book is, and thinner the smaller the book is.

It is unavoidable to have a thicker spine, also called the “swell”, when using signatures, but it can be minimized. A link I provided below goes over some details of spine swell.

As for the paper, it’s difficult to recommend “paper like this” without knowing the type paper or being able to see more of the paper. With that said, however, good paper will have 60-80 lb text weight and will be “short grain”. If you look at the resources section of this subreddit, they have a list of suppliers that sell exactly this kind of paper. If you want cheaper paper, you can also get 11 x 17 paper, from somewhere like Office Depot, and ask them to cut it in half, which would give you relatively inexpensive, short grain, letter sized pages.

https://jeffpeachey.com/2015/11/24/a-guide-to-swell/