If laying vinyl the floor doesn’t necessarily need to be level,if it doesn’t catch your eye u should be fine when ever per preparing a sub floor I would treat it with Dpm primer and finish with a 6mm wbp plyboard a good floor layer will always screed the joints and sand for preparation

i dont know any qualified lvt fitter that would install on someone elses prep work, best get a installer to look at the job as he will have the correct training and meters etc to install correctly

As much tile adhesive should be removed as possible. Any remaining needs to be fixed solid to the subfloor, presuming that is concrete. Being a fairly modern house it should have a dpc and therefore be dry. It could be checked using a digital hygrometer, which we have available, and to smooth use either a primer and water-based self leveller from a reputable source, not diy store, or use Arditex NA smoothing compound, which requires no priming.
Good to go, whatever you decide to lay. Happy to help.

It would be much better to either even out the floor or by putting 6mm ply down first. This doesnt solve the problem but it certainly makes the uneven floor less noticed