I was told that when the controller of the creature changes it exits the battlefield and enters on the other side of the playing field on the new controllers side. If what you’re saying is true does that mean it keeps all previous attachments?

You were told wrong. There are no “sides” of the battlefield; there is just the battlefield, it’s all one zone. Changing controllers does not count as changing zones.

Any cards that were attached to the creature before it changed controllers will still be attached (assuming none of them specify that they must or can’t enchant a certain player’s creatures).

This would play out as follows:

You cast Curse of the Swine targeting several creatures.

Dack’s Emblem’s ability triggers, going on the stack ahead of Curse of the Swine.

Assuming no other responses, the Emblem’s trigger resolves first, and you gain control of the targeted creatures. Remember: Spells and abilities on the stack resolve in ‘First In, Last Out’ order. (If you’re confused about how that works, check out the very first MTG Tutorials: What is the Stack?)

Finally, Curse of the Swine resolves. The targets have not left the battlefield, and Curse of the Swine doesn’t specify that it has to be creatures your opponent controls, so the targets are all still legal. Remember: Changing controllers is not the same as changing zones. Your newly stolen creatures are exiled by Curse of the Swine, and you -- as their controller -- get that many 2/2 Boar tokens.

Supreme Verdict doesn’t target, so it won’t trigger Dack’s emblem’s ability. (Cards that target use the word “target”.)

Mana Reflection is a replacement effect. It modifies how much mana is added to your pool each time any mana is added (assuming it came from tapping a permanent).

Doubling Cube’s ability is an activated ability. It doubles how much mana is currently in your pool at the specific time it is activated.

If you’re not sure what the difference is, check out:

MTG Tutorials #2: All About Activated Abilities

MTG Tutorials #24: All About Replacement Effects

While you’re at it, you also might want to check out:

MTG Tutorials: Extra Credit — All About Mana

Up to speed? Good. 

Here’s an example of how these two cards play together:

Say you control Mana Reflection, six Forests, and Doubling Cube. You tap all six Forests to add 6 green mana to your mana pool, but Mana Reflection’s replacement effect doubles that 6 mana to 12. Then you spend 3 mana (putting you down to 9 in your pool) to activate Doubling Cube’s ability. 

When that resolves, Doubling Cube sees you have 9 green mana in your pool currently, and tries to double it by adding 9 more — but wait! Mana Reflection sees you trying to add mana to your pool again, and doubles the amount being added. Because of Mana Reflection, the 9 mana from Doubling Cube becomes 18, which gets added to the 9 you already had.

In the end, you’ll have 27 green mana floating in your mana pool — not bad considering it all started with six measly Forests.