Wow, what a lot of questions. I'll have a go at answering the one that I feel most competent about (or at least have the strongest conviction).
Mellow sounding tenor. My view is that in 40 years of playing I've never experienced "mellowness" in the horn itself. There have been a few attempts on SOTW to put percentages on the contribution of the various components to the overall sound. I hold the view that the biggest contributor which, to some extent you're stuck with, is your own mouth/teeth/jaw. The next biggest thing is the mouthpiece/reed and here you get the biggest variability: small changes in the structure give large changes in sound. Then comes the neck, but you can spend a lot of money here for no discernible difference to the sound. Last comes the body where the important things have to do with the comfort and playability although horns differ in their internal geometry and this can affect timbre.
I'm glad to hear you're playing in a band - it's the best incentive for improvement that there is.
As for getting what you want at the price you're prepared to pay, well there are some great bargains to be had and there are some shocking rip offs. It's entirely possible that you'll find your dream sax first off - it happens. But it's more likely you'll start down the path of finding bits of what you like in a number of horns that pass through your hands.
I should be able to find a used horn that plays in tune and that is in reasonable mechanical condition for the price you mention. Once you have that, you can make a hell of a lot of progress toward the sound you want by practice, listening and mouthpiece/reed selection. I should add that I'm not advocating wide experimentation with mouthpieces yet.. That can be a slippery slope. Keep playing and playing that Link until you're convinced that it's holding you back. Then borrow some other pieces to try.
Mellow sounding tenor. My view is that in 40 years of playing I've never experienced "mellowness" in the horn itself. There have been a few attempts on SOTW to put percentages on the contribution of the various components to the overall sound. I hold the view that the biggest contributor which, to some extent you're stuck with, is your own mouth/teeth/jaw. The next biggest thing is the mouthpiece/reed and here you get the biggest variability: small changes in the structure give large changes in sound. Then comes the neck, but you can spend a lot of money here for no discernible difference to the sound. Last comes the body where the important things have to do with the comfort and playability although horns differ in their internal geometry and this can affect timbre.
I'm glad to hear you're playing in a band - it's the best incentive for improvement that there is.
As for getting what you want at the price you're prepared to pay, well there are some great bargains to be had and there are some shocking rip offs. It's entirely possible that you'll find your dream sax first off - it happens. But it's more likely you'll start down the path of finding bits of what you like in a number of horns that pass through your hands.
I should be able to find a used horn that plays in tune and that is in reasonable mechanical condition for the price you mention. Once you have that, you can make a hell of a lot of progress toward the sound you want by practice, listening and mouthpiece/reed selection. I should add that I'm not advocating wide experimentation with mouthpieces yet.. That can be a slippery slope. Keep playing and playing that Link until you're convinced that it's holding you back. Then borrow some other pieces to try.