Yes and yes.

As you have practically embedded every relevant rule, I will only give the reasoning without citing the sources.
Spiked Chain Training makes it double weapon, double weapon means 2 one-handed weapons, one in each hand, and you can attach those. Embedding only works with some specific weapons, and the spiked chain is not one of those. It is not hidden, as only embedded components can be hidden.
Imagine it as a scarf, extending from one hand to the other over your neck.

Houseruling it so that it is embedded and hidden would not be overpowered however in my opinion, so talk to your DM.

Do not forget to add Dual Implement Spellcaster to your Eldrich Stike, as you make an arcane attack while wielding an implement in each hand.

There is one catch however, you have to enchant the item yourself, as a Pact Blade has to be a light blade, and Spiked Chain Training does not make Spiked Chain a light blade. You can enchant it, as you treat it as a light blade.

Improvised Weapons are not a subset of Melee or Ranged Weapons, but they are Weapons.

Let's begin with the identification of the Improvised Weapon as a Weapon (and not an object or Armor, or anything else). Jeremy Crawford supports this:

An improvised weapon is, indeed, a weapon, but only the moment it's used as such. A chair/shield/etc isn't a weapon otherwise.

What type of weapon is it? An Improvised Weapon

Chapter 5 of the PHB (page 149) contains a list of common Melee and Ranged Weapons. That is the list for things that require Melee Weapon or Ranged Weapon attacks. Existing items within the ruleset that are not listed here would likely not qualify, but items that are not listed which have similarities would (i.e. katana as longsword). A shield is not on the list, but is under Armor.

Whether or not Improvised Weapons are on the list is the exact question being asked here.

Jeremy Crawford takes a bit about the object vs process here:

In D&D, a weapon is an object. A weapon attack is a process. Sometimes the rules let you use nonweapons to engage in that process

An Improvised Weapon is neither a Melee Weapon nor a Ranged Weapon (from the list previously defined on page 165.) It is it's own category that can be found pages 147-8 in the PHB.

Type of Weapon is different from type of Attack

While you may make a Melee or Ranged Attack with one, it does not turn the Improvised Weapon into a Melee or Ranged Weapon. It is just a Melee or Ranged Attack that is made with an Improvised Weapon.

Crawford covers this in the following exchange:

A melee weapon attack is a melee attack with a weapon.

and the following exchange with a user (the first quote):

@Sebkha Not to be confused with: an attack with a melee weapon.

@JeremyECrawford That's correct, since an attack with a melee weapon can be a ranged attack if you throw it

Proficiency points the way towards a different Type than Melee/Ranged

If an Improvised Weapon is ALWAYS either a Simple or Martial Melee or Ranged weapon, then there would have been no need for this clarification on proficiency in the Improvised Weapons rules:

In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.

If it was a given that an Improvised Weapon was always a Simple or Martial Melee/Ranged weapon, then proficiency would always apply. In addition, the only way to gain proficiency in an Improvised Weapon is through the Tavern Brawler feat. This would be unnecessary if Improvised Weapons were classed as part of Melee/Ranged as your proficiency would already be granted if that was covered by your race/class.

Crawford discusses this a bit here:

A shield certainly isn't a weapon, but like many things, it can be used as an improvised weapon.

There is clearly a differentiation in Crawford's eye between Weapons and Improvised Weapons.