There are three issues here, I think: Keywords, the two different kinds of proficiency, and permission by omission.
But before I go into those, a word: As always there are explicit features/feats/enchantments which break the rules, and that's why we call D&D an "exception-based" system: it deals in rules which apply universally unless (until) exceptions are made, so there is no need to enumerate the possible exceptions. We simply assume the rule unless told otherwise in a particular instance.
Keywords
If a power has the weapon keyword, and only if the power has the weapon keyword, does a weapon enchantment (enhancement bonuses and other features) apply to that power. Ditto with the implement keyword and implement enchantments.
Proficiency and the Proficiency Bonus
"Proficiency" means that you've had training in the use of a weapon or implement, but mechanically it means totally different things whether you're talking about a weapon or an implement.
Weapon Proficiency and the Proficiency Bonus
Proficiency with a weapon means that you can add that weapon's "proficiency bonus" to attack rolls. Only weapons have proficiency bonuses, they only apply to powers with the weapon keyword, and they have nothing to do with whether enhancement bonuses can be applied (see below for that bit).
Implements, Enhancement Bonuses, and Permission by Omission
You need to be proficient with an implement in order to add its enhancement bonus to attacks and damage with implement powers. You do not need to be proficient with a weapon in order to add its enhancement bonus to attacks and damage with weapon powers, but you don't get its proficiency bonus to the attack roll. (In either case, you can only add the enhancement bonus of one item at a time to an attack unless you have a rules exception which says otherwise.)
I arrived at this conclusion because the magic implement rules say you need to be proficient for the enhancement bonus, but the magic weapon rules don't. Permission by omission is sloppy, but has solid precedent.
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.
Best Answer
Double weapons are interesting. For all purposes you treat them as two separate weapons. You don't get any extra attacks with them. Both ends are enchanted with the same enchantment and other than flavor and cost efficiency they function as two separate weapons.
The big bonus to dual weapons is in cost efficiency. A two weapon ranger or other build that dual wields needs to have two magic weapons to be completely efficient, this means either using a lesser enchanted weapon in one hand or forgoing magic armor or neck slot item (unless your DM is generous with the magic item or gold drops). Using a double weapon allows you to have the same enchantment on both ends of a weapon without having to collect two magic items to do so.
As a cleric you won't get much use out of a double weapon unless you have a feat or power something that allows you to add both of your implement's enhancement bonuses to your powers. Note, that using a staff a dual weapon should prevent you from dual wielding staffs as all dual weapons are two handed (unless I am mistaken here).