human biology - How are non-glucose sugars metabolized in the body?

In my biology book's section on disaccharide metabolism and glycolysis, it states that sugars other than glucose must be acted upon to enter glycolysis. Let's take sucrose as an example. Sucrose is hydrolyzed in the small intestine by sucrase. The resulting fructose and glucose are absorbed and transported to the liver via the portal vein. My question concerns the fate of fructose.

To undergo glycolysis, the book states that fructose is converted into either fructose-6-phosphate (F6P) or fructose-1-phosphate (F1P). Let's say it is converted to F1P. Aldolase splits this into dihydroxyacetone phosphate and D-glyceraldehyde. Triose kinase then converts D-glyceraldehyde to glyceraldehyde-3-phosphate, a glycolytic intermediate. Where is this occurring in the body? Are we still in the liver? I can't imagine that all the fructose we consume is undergoing glycolysis in the liver. To leave the liver as a sugar, it would have had to been converted to glucose, right?

In classes I've taken, I've been told that sugars that enter the liver are pretty much all converted to glucose. Once they are converted to glucose, they can be distributed to the rest of the body, stored as glycogen, etc. If we are going straight from fructose to F1P to a glycolytic intermediate, we couldn't have left the liver. How is such a transformation even useful? Anyone care to shed some light on this?