A solid grap of mathematics at an advanced level is certainly necessary, if not sufficient, to become an engineer. One of the biggest class of people inclined to become engineers who never make it (call it the "Popular Mechanics crowd") are people who are mechanically inclined, but simply can't make it through vector calculus and differential equations.
Dexterity is probably less important than it was half a century ago, now that CAD has become ubiquitous, we use computers instead of slide rules to do calculations, and even modeling can be done with "3-D printers". Consider technical drawing, e.g., which is one of the most dexterity intensive things engineers do (much of the actual assembly is done by technicians). If you can trace a drawing, you have the manual dexterity to draw it, the rest is conceptual. Certainly, dexterity can be useful, but it isn't make or break, within reason.
Creativity is another ability which can be a plus, but isn't necessary for everyone. Sure, there are glamour jobs in engineering designing new inventions or remarkable structures. But, there are also no shortages of jobs that call for straightforward application of proven engineering principles to routine situations in an unimaginative way. Somebody has to design highway overpasses in Kansas, supervise the design of neighborhood transformer stations in growing suburbs, rework the hydraulics on a cargo lift for a new model delivery truck which rides six more inches off the ground than the last model, and figure out the optimal amount of artificial food coloring required to make the new purple version of Mountain Dew.
Also, as noted above, engineering skills often have non-engineering applications which pay better than straight engineering jobs. For example, I have a cousin who is a chemical engineer, but has spend a decade or so as a customer service/sale representative for a company that sells sophisticated components for technical systems. She spends most of her time helping her engineer customers figure out which parts they really need and how many of them they need.
This isn't to say that a good mechanical sense (psychologists sometimes call it "structural vision") isn't a good and necessary thing, and that there isn't room for a creative engineer whose good with his hands, but plenty of people make a good living without either, and plenty of creative people who are good with their hands find outlets that are more satisfying than engineering for them.