Is it possible to share the character’s profile through a web page

If I understand it properly, I think there's a problem in the question itself. As you said:

The merchants give you Scroll Fragments for white items, so we can sort of divine that that is the base item value. But it doesn't really flow well from there [...]

It doesn't add up, indeed, and you gave the exact reason in your own question:

• Path of Exile has done away with a single unified currency entirely. Exact
• [...] there's no way to tell what anything is worth, or which items are worth more as part of the local economy. More or less correct

Actually, while there isn't any curve to express the value of the items in terms of value, we probably could figure one out based on the difficulty to obtain the item from the vendors.

• You can get a Scroll Fragment with pretty much any crap item (which are easy to find).
• You can make a Scroll of Wisdom with 5 Scroll Fragments (which are easy to find, but you need 5 of those).
• You can get a Portal Scroll with Scroll Fragments (but you need 15 of those). It's getting harder to obtain, but still feasible.
• You can get an Armorer's Scrap with an piece of armour with a 20% quality, or with several pieces of armor with a total of 40% quality: feasible, but you'll need patience.
• You can get a Chromatic Orb with any item that has 3 sockets of different colors and linked together: you'll have to be very patient to find one of these.
• You can get an Orb of Chance with 2 rare items that share the exact same name: it's getting super hard.
• Etc.

With that in mind, the whole currency thing becomes a matter of recipes, and the "value" of each currency item is determined by its difficulty to be "cooked" (let's put it this way) and, therefore, its rarity.

The vendor recipe system allows the player to sell items to any town vendor in exchange for a multitude of currency items and equipment. Each recipe requires semi-specific items or combinations of items be put into the sell window at the same time, and the outcome will change based on any recipes that have been matched.

Source and list of known recipes: http://pathofexile.gamepedia.com/Vendor_recipe_system

The only remaining uncertainty in this system is the drop rate of each currency item (because, of course, it's also possible to obtain them by killing mobs). But I couldn't find any information about it.

The dev diary given by Gnoupi is also an excellent reading: Dev Diary: Rethinking Gold as a Currency.

tl;dr: Items that people want to have are expensive, other items are not. I suggest using a service like poe.xyz.is to get a feeling for the value of an item you might want to sell. Especially for uniques it can give you a very good idea of an item's value.

This is a pretty open-ended question so a conclusive answer may be hard to come by. That said...

Affixes come in different brackets, or tiers. For example, % Increased Physical Damage comes in 7 tiers ranging from "Heavy" (+20..49% IPD) to "Tyrannical" (+150..169% IPD). The stronger brackets are of course more valuable, while the weaker ones may even reduce the value of an items since they prevent stronger ones from spawning.

Movement speed is incredibly important on boots since it cannot spawn on other gear, and it's rare to see a pair without MS sell at all, let alone fetching a high price.

Affixes that align well with the base item type are more expensive. E.g. Increased Critical Strike Chance is much better on a Dagger than it is on a Maul, because Daggers already have a high base crit chance and are therefore commonly used in crit builds, while Mauls are strength-aligned and often used by characters who cannot crit due to Resolute Technique.

Affixes that align well with each other can increase the value of an item - or rather, affixes that don't align can relegate an item to the vendors. A combination like % Increased Physical Damage and flat Increased Physical Damage, or +2 to Fire gems and +1 to all gems can get quite expensive, while no build I know of makes good use of both Attack Speed and Cast Speed. Especially on base types with a large mod pool (e.g. jewellery, daggers, wands, sceptres, staves) having well-aligned affixes can increase the value dramatically.

Resistances are pretty popular on socket-less items because they're easier to swap around to accommodate changes in your other gear.

With the release of Forsaken Masters, a 5-affix item may well be more valuable than that same item with a 'decent' 6th affix, because it leaves some room for the buyer to customize the item.

Corrupted items are in general less valuable than their non-corrupted variants because the sockets are locked, but if they rolled a very good corruption this is of course not the case.

It could be said that different factors have multiplicative scaling - e.g. a 6L item is expensive, and an item with good affixes is expensive, but a 6L item with good affixes will not simply be twice as expensive - it will cost you three and a half fortunes.