Why doesn't water react with the aldehyde to form a linear geminal diol?
It does, and depending on the nature of the aldehyde, this will be the major species. You are right in saying that the high concentration of water helps to form the diol. In other solvents, the diol form will be present at lower fraction, even when they contain a bit of water. The Wikipedia article on geminal diols has some equilibrium constants for various aldehydes and ketones in aqueous solution: https://en.wikipedia.org/wiki/Geminal_diol#Reactions
Glucose's overwhelming preference as a cyclic pyranose is commonly explained by the cyclization being intramolecular; while the aldehyde group in its linear form can react with other molecules like protein and lipids during rare encounters, it far more readily reacts with an alcohol group that is always around by virtue of being on the same molecule.
Yes, that is the special thing about glucose: the aldehyde can form an intramolecular hemiacetal without strain. It can also form a geminal diol, but that is no different from other aldehydes, so it is usually not mentioned in that context.
Is the geminal diol ever important for the reactivity?
There is an enzyme called aryl sulfatase which has an aldehyde (a cysteine modified post-translationally to a formyl glycine) in its active site [review]. X-ray crystal structures show that the main form is the hydrated species, i.e. the geminal diol. In the suggested mechanism, one hydroxyl attacks a sulfate ester, displacing the alcohol. Then, the second hydryoxyl group reforms the aldehyde and releases the sulfate in an elimination reaction.
Are there other instances where we don't explicitly mention reaction with water?
The hydrogen ion occurs in aqueous solution in various hydrated states. Carbon dioxide in aqueous solution occurs either hydrated (carbonic acid) or not. In either case, it is easier to lump the species into one than to talk about them separately.
What is the equilibrium constant for the hydration of the aldehyde in glucose
I did not find data for glucose, but did find some for glyeraldehyde 3-phosphate (Petersson and Petersson 1999). [In the HTML version of the paper, some equations and schemes don't show, but the PDF is an improvement.] In aqueous solution, only about 3% of this phosphorylated sugar is present in the aldehyde form. In the reaction they were studying, only the aldehyde form reacts in the enzymatic reaction, so there is a slow non-enzymatic first step to dehydrate the diol and a fast second step.
Wikipedia has an article on glycoaldehyde, with 4% of it in the aldehyde form.
If glucose behaves in a similar manner, most of the linear form would be in the diol form. Equilibrium constants given in the literature probably use the total concentration of linear glucose (hydrated or not) in the equilibrium constant expression rather than the actual concentration of aldehyde.