You often can't tell by looking — stud walls routinely carry roof struts or floor joists, and solid walls are not automatically structural. Useful clues: what runs above and below the wall, joist direction, and whether the wall continues on the floor above. But the honest answer is that an engineer confirms it in one short inspection, which costs far less than discovering the truth mid-demolition.
Yes. Removing or opening up a load-bearing wall is building-notifiable work: you need engineer's calculations for the beam, a building control application, and an inspection before the steel is boxed in, leading to a completion certificate. If the beam bears into a wall shared with a neighbour, the Party Wall Act applies as well — the panel coordinates both.
It is fixable. An engineer inspects what was done, checks (or designs) the support arrangement, and produces the calculations needed for a regularisation application to building control. Buyers' surveyors flag unauthorised removals constantly, so regularising before you sell — rather than negotiating a price cut under time pressure — is almost always the cheaper route.