For me, malty is one of those kind of 'irreducible' qualities, i.e. it's hard to describe exactly what else it tastes like besides malt, in the same way that it's hard to say exactly what 'grape-y' tastes like, aside from 'like grapes'.
A very large part of maltiness as a distinct flavor is melanoidins, the product of Maillard reactions between sugars and amino acids during malting. Because this reaction is driven by heat there's a good correlation between the color of the malt and the intensity of the malty flavor (at least until you move into the dark crystal/chocolate/roasted malt range). Well-modified malts offer higher levels of sugars and amino acids before kilning, and therefore tend to be maltier. Vienna, Munich, and medium crystal malts are good examples of higher melanoidin malts, although all lighter malts also get their malty flavor almost entirely from melanoidins.
Caramelization is another factor, though it is virtually non-existent in many malts. Sugars except fructose caramelize only above 320°F (fructose, above 230°F). As an example, dark Munich malt doesn't usually get above 220°F during kilning so it and any lighter malt should contain no measurable caramelized sugars. It becomes a factor in darker malts like Brown malt, darker crystal malts and chocolate and roasted malts, where kilning temperatures reach well beyond the point of caramelization. Maillard reactions can continue at this temperature too (though it's often too dry), so the balance of the two dictates the characteristic of the particular malt.
Because the formation of trademark malty flavors will always depend on the spectrum of sugars and amino acids (as well as many other non-melanoidin-forming compounds) in the germinating grain when it's kilned, it's not necessarily a flavor that has many true analogues in every-day life. The most common synonyms I've heard are bready, toasty and bisciuty. These examples no doubt share many of the same flavor compounds.
To address a few of your questions directly:
Is "malty" the same as "body"?
Not really. Body's much more about mouth-feel, the perceived thickness or substance of a beer, whereas malty is a distinct flavor.
Is it possible to have a low body but malty brew?
Yes, definitely. Think of styles like English milds and bitters, some darker Belgians, American red ales, etc. Malty flavors, but no cloying fullness.
which malts ... contribute a "non-malty" flavor?
Strictly speaking, none do, since all malt intrinsically has the flavor of itself. Apples always taste like apples, even if they don't taste like all other apples. However, lighter malts will have less intense malt flavor than darker ones.
what is the word for non-malty?
Sweetness is tricky because there are so many uses of the term which do not imply sugary sweetness (literally the taste of fructose/sucrose). In this sense, sweetness, as we most often use the word, is very closely related to body, as both have much to do with the perception of the thickness and mouth-feel of a beer. In my experience, the best prediction of sweetness (as you suspected) is the final gravity.
Of course there can be actual, sugary sweetness in beers. Some fermentable sugars may survive fermentation, for various reasons. Some unfermentable sugars can have perceivable, if only very subtle, sweetness. Dextrins are virtually flavorless, and I've read multiple authors suggest they are not a high enough molecular weight to account for the body of a beer alone, though they will contribute somewhat. Some dextrins, as Pepi suggests, may be further reduced to simple sugars by salivary amylase, though this is probably limited in scope since the enzymes in saliva (α-amylase, virtually identical to that in malt) are incapable of hydrolyzing the branch-points of dextrins, meaning only dextrins larger than those derived from a typical mash would yield much sweetness. They could be a factor in very low diastatic-power mashes, which may leave such larger dextrins unconverted in the wort which pass undermented into beer. A well-planned (short and hot) mash schedule could easily promote this kind of sweetness.
Bitterness, as you suggest, is also a big factor in the perception of sweetness, in that it tends to offset it. Thinking of bitterness and gravity as ratios rather than isolated numbers always made more sense to me. It makes it very easy to 'scale' a beer of a particular sweet/bitter balance to a higher or lower ABV%; in my experience maintaining the ratio maintains the balance, too.
Many other flavor compounds in beers can be perceived as sweet, either directly or in their contribution to a beer's body. Alcohols (both ethanol and fusel) contribute a characteristic mouth-coating fullness, especially at high levels. Esters may have a complementary effect. Too much diacetyl lends its own particularly slickness, and various proteins that survive into the beer contributes viscosity and body, much more so than dextrins. Unfermentable melanoidins and caramelization products will also boost body and sweetness perception.