Hinges: Much depends on what type you intend using, but if they are close to a corner, you can simply drill into the edge and let in a dowel at right angles to the hinge screws.
I do it in softwood when I need to put a screw into end grain, and it works well. On a jewellery box scale, I'd probably use a Mahogany-ish hardwood because of the fine pitch of the screws, but once cut off flush with the side there is very little chance of it telegraphing through your veneer, as it will move across its width and only by a tiny amount.
Furthermore, most MDF has a harder "crust" on the flat faces, and is a bit lighter density in the middle of the board thickness, so again there is a slight room for movement (depending on thickness). I think it's more pronounced in MR MDF.
For what it's worth, the author of one veneering primer I have strongly recommends MDF as a ground for veneering projects, because of its stability. The book itself isn't brilliant as a primer (so not wholeheartedly recommended), but his work (and Brian's!) proves that MDF is a good choice.
I also watch a lot of Thomas Johnson's stuff on YouTube. He's an American furniture restorer, and a huge amount of his work seems to be repairing veneered work that's been done on a natural wood ground, which in turn has shrunk in a modern home. Some of the damage - shrinking and splitting is rather soul-destroying to see (but he regularly seems to manage the impossible!). MDF does move about, very slightly, but far less than natural materials.