No. Nein, Nyet. Just in case you missed it, NO.
+P is a way of denoting cartridges which are loaded to a higher gas-pressure than what is considered normal for that cartridge.
For all intents and purposes 'Special' should just be held as a name. In some cases when a manufacturer would bring out a new cartridge firing a bullet diameter shared by another cartridge they would call it a 'Special' to distinguish it, but that is a much longer discussion to cite the various and sometimes contradicting methods in which the term is applied.
In the case of something like a .38 Special; the +P version would be .38 Special +P; indicating a higher-pressure loading of the .38 Special cartridge.
Not all guns are created equal and while most modern-production guns can safely use +P ammo (at the cost of increased wear on the gun) this is not a universal rule; and you should never use +P ammo in a gun if there is any question of whether or not it can handle the higher pressures.
Same thing. Yes.
+P ammunition differs only in having more powder - the cartridge dimensions remain the same. Most Colt products are capable of handling +P ammunition, and even +P+ - however, you should contact customer service with the model and serial number of your firearm to get confirmation directly from the company that your pistol is rated to withstand the additional pressure generated by +P ammunition.
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If it is only listed as .38 Special (spl), use only that ammo. If it is .38 S&W special, use only that ammo. Don't try to mix them. The diameters are not the same. If you still cannot determine the actual cartridge caliber, take it to a gunsmith and have them identify it for you.
No, just use same ammunition
Question is too broad to answer. In general, use ammunition that is the same as the caliber or gauge marked on the firearm. If it is marked .38 Special, use only .38 Special- NOT .38 S&W, NOT .380. NOT .38 Super.
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Due to a special property of zero, x+0 is the same as x.
That depends on what the categories are.
No. They have matching dimensions, but differ in other areas, such as neck angles. .223 ammunition can be safely fired in a 5.56x45mm chamber; 5.56x45 ammunition is not recommended to be used in a .223 Remington chamber.
This is very complicated and depends mostly on the barrel lengths of the pistols and the specific loadings of ammunition which are being compared. It is possible to buy or handload ammunition such that either of these are more powerful than the other or that they are the same. However, it is possible to load .38 special with heavier projectiles than a typical .380 auto load and stay within safe loading perameters. Understand that total energy is a function of projectile weight and velocity. Lighter projectiles with the same charge of gunpowder will likely have much higher velocity than a heavier projectile.