Are you sure the bench you purchased has decline ability? In looking at the Amazon page, it doesn't seem like it. You could but the bench at a minimal decline by placing somethig sturdy like a board...
Soreness isn't necessarily an indicator of a good workout. My first thought would be to check your form, which is particularly important with shoulder movements like lateral raises. There's often a...
What does your "back auxiliary" day typically consist of? I'd say avoid excessive movements which target the shoulders etc. if you're planning on trying to PR tomorrow. If it's just a regular chest...
I enjoy using a diverse range of equipment...stuff that I wouldn't be able to have in my home unless I had a traveling circus tent gym like the Rock. So, I vote for the gym.
Don't worry about bulking up your waist. Focus on your back, shoulders and chest - broadening this area will allow you to have a great V-taper. Target your core with regular volume.