“sealing compound” for attaching a sink

You say the supply line "upper border" is touching the sink porcelain... Does that mean that if you raised the new sink an inch or two the porcelain would clear the supply lines? If so I'd go that route. I know many IKEA vanities stand on legs; if that's the case with yours perhaps you could replace them with taller legs, even if you have to make them yourself. If it's not on legs the vanity can be shimmed up and the shims hidden with base molding. As @shirlock mentioned pics of what you're installing would be helpful.

And as for the mounting studs, if you don't end up needing to open the wall, and as long as they're going to be covered by the new sink, I'd cut them off with an angle grinder as you suggested.

This is not a trivial project, as apron front sinks have straight edges on the front sides and bottom.

2 cleats should span under the sink to support the weight. They should be sturdy enough to make the cradle, perhaps 1x3 or 1x4, oriented like joists. You could reinforce the cabinet sides with the same size boards laid flat and extending to the cabinet bottom (like posts). Using two pocket screws on each end of the joist would give you a bit of adjustability. Just don't drive them home until you're happy with the positioning.

The side cuts and top cuts should be back cut to make final shaping easier. Making a foam template will be helpful and easier than sanding the top. Blue or pink EPS foam at 1 inch should fit your needs. A jig saw, belt sander, and perhaps a rotary tool ( like Dremel) will be most useful.

Final affixing of bowl to top can be done by setting in a bead of sealant-caulk along top and front and perhaps a dot on each joist.