First of all, the budget is not of concern.
My needs: able to walk and carry the bike, able to clip out easy (clip in easy is a bonus but not necessary), pedals/cleats must handle a moderate amount of mud, weight is a semi-concern. The shoes must also be able to be used as normal shoes, like walking in a city or going into a bar something like that, so I don't need to carry a second pair of city shoes on my bike. I think that's about it for the needs, again don't think of the price, just think of the best gear for the mentioned needs. Thanks a lot.
In a nutshell: what's the best combo for adventures rides on non-paved roads (gravel or easy trail) for hundreds of km, with also being able to use the shoes as casual shoes.
Just to give a simple answer. Look for comfort. So that means in shoes, they fit your feet, and don't discount the option of a bike-fitter fitting insoles in your shoe. Then also let the bikefitter set up your cleats, this will most likely make the biggest difference. In terms of shoes, the fit has to be good (Lake shoes has 4/5 different fits in each size,so that may be an option to try out various of their models). And of course don't choose the stiffest soles if you want comfort, also on those moments you are off the bike and still wearing those shoes. So carbon soles yes, stiffest carbon soles no. In terms of pedals your knees and ankles likely don't enjoy being tightly locked over such long time, and would want to move a bit freely at least. So look for pedals with at least 6D of float. I think Look x-track race is a good options, or even the en-rage option if you use flat shoes. They have a slightly bigger platform compared to most systems which would support your feet.