This actually does work together. Note the text:
You gain a +1 feat bonus to attack rolls you make with any weapon with which you have proficiency and with a wand or another item designated as a bard implement.
The first part gives you a bonus to attack rolls you make with any weapon with which you have proficiency. The second gives you bonuses with wands or other bard implements.
Note that it doesn't say "weapon attack rolls." Instead, it says "attack rolls you make with any weapon." They're not the same.
Now, the rules for Weapons as Implements and vice versa:
Using a Weapon as an Implement: If an adventurer is able to use a weapon as an implement, the weapon works like a normal implement, but the adventurer uses neither the weapon’s proficiency bonus nor its nonmagical weapon properties with his or her implement powers....
When an adventurer uses a magic version of the weapon as an implement, he or she can use the magic weapon’s enhancement bonus, critical hit effects, properties, and powers. However, some magic weapons have properties and powers that work only with weapon powers (Rules Compendium 275).
You're making an attack with a weapon with which you have proficiency. You happen to be using it as an implement, but that doesn't matter- the text says you get the bonus with attack rolls made with any weapon, and you're making an attack roll with a weapon.
This issue comes up in a variety of ways in the game. For example, if you're using a Frost Shortbow, you could attach a Siberys Shard of Merciless Cold to it to do extra damage, which would work even when making an implement attack. If you're using a Weapon of Speed Shortbow, you make make a Ranged Basic Attack with it once per encounter as a minor action...even if the RBA is using the bow as an implement, not as a weapon.
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
There is a weapon proficiency feat; it grants you proficiency with any one weapon of your choice, regardless of what kind of weapon (simple/martial/superior) the weapon is.
If you use a weapon you are not proficient with then you do not receive its proficiency bonus to attack rolls (the character builder is buggy about this and will often show you receiving it, but the rules clearly state that you don't get the bonus if you're not proficient). Nothing reduces this penalty; you could be proficient with every single weapon except one, and you would still not get the bonus for that one weapon.