Squeezing juice out of pickles and other foods saturated with external liquids:
A. Squeezing out the liquid not for the sake of adding the liquid into food:
For the sake of drinking their liquids: Fruits and vegetables that are pickled in salt and vinegar or in salt water, as well as those which are cooked, are forbidden according to all opinions to be squeezed for theirs liquids.
The above is only a Rabbinical prohibition: If one [transgressed and] squeezed [the above foods] for their liquids he is not Biblically liable because even though the liquids were fully fledged liquids prior to becoming absorbed into the [foods], such as [occurs] if one pickled or cooked them in water or wine and vinegar and the like of other liquids, nevertheless since now they are being squeezed from food they have the same status as food and it is thus like separating food from food. Nevertheless it is Rabbinically forbidden to squeeze them for the sake of their liquid since they were fully fledged liquids prior to this. However raw fruits since their squeezed liquids never had upon them the status of liquids [prior to the squeezing] therefore it is allowed to squeeze them according to those opinions which permit doing so.
In order to enhance the food being squeezed: One is allowed to squeeze pickles and the like for their own need, in order to prepare them to be eaten through doing this, [such as to making it taste better through doing so].
The reason for this allowance: This case is not prohibited due to “detaching” since one does not at all need the item being detached, which is their juice.
B. Squeezing cooked and pickled foods into food:
It is permitted to squeeze pickled and cooked [foods] for the sake of their liquids into a pot of food in order to enhance it. However if the pot does not yet have food in it then it is forbidden to squeeze into it even though one will eventually mix food into it, as was explained above [in Halacha 3]
C. Squeezing out the food part of a fruit or cooked food 
Apples or other cooked fruits are allowed to be squeezed in order to remove from it all of its content because he is squeezing out all the food from it and is not [doing so] in order to take out its liquid.
Summary: Squeezing juice out of pickles and other foods saturated with external liquids:
Squeezing them in order to drink the liquid: Is Rabbinically forbidden
In order to enhance the food being squeezed: Permitted
In order to place into food: It is permitted to squeeze it directly into food. However to squeeze it into an empty vessel is forbidden even if one plans to place it into food.
Squeezing out the entire content of the food: Such as to squeeze out cooked apples from their peel is allowed.
Why does squeezing liquid out from a food not contain the Borer/separating prohibition?
As the food and liquid that is absorbed in it are considered one food to which the separating restrictions do not apply as explained in The Laws of Borer Chapter 1.
If a food will only become edible due the squeezing may it nevertheless be squeezed?
No, as this poses a Borer prohibition. This may not be done in any cases, whether with intent to squeeze the liquid into food or to sweeten it.
May one swipe up the gravy on his plate through pressing bread against the floor of the plate?
If it is not for certain that liquid will be squeezed out of the bread in the process, then this is allowed. However if in the process of the pressing liquid will be squeezed out, it is forbidden.
May one squeeze liquid out from his chalah if it fell into liquid?
Seemingly yes as it is in order to enhance the food.
May juice which had been directly squeezed into a food be then squeezed out of that food?
- For example: One squeezed lemon juice into his salad and then desires to squeeze it out after it having absorbed the taste?
This is allowed to be done as since the juice which was squeezed into it never received a liquid status it therefore does not receive the same status as pickles and the like which contain juice that was previously defined as liquid. Nevertheless, it is proper to only rely on this in a time of need.
When squeezing out liquid in order to enhance the food may one squeeze the liquid into a plate and then drink it?
Although there are Poskim which are lenient in this, the Tzemach Tzedek rules one is not to be lenient to do so. Practically one is not even to squeeze the liquid into a plate but rather he is to squeeze it in an area that it will go directly to waste.
Practically, may one squeeze out the oil from a Sufganiya/ Latke? 
If he is doing so in order to enhance the food and make it more edible then it is allowed. This includes if he is doing so due to that the oil absorbed in the food bothers him. However if one desires to use the liquid for something then it is forbidden. As well to do so for no reason at all is to be avoided, as explained above.
If one is doing so in order to both enhance the food and reuse the liquid, then see above Q&A where this matter is disputed, with the Tzemach Tzedek ruling that it is forbidden.
May one squeeze out the oil from eggplant or vinegar from cucumber salad?
Has the same laws squeezing oil from Sufganiyot, explained above.
 Why is the fish gravy any different than juice of a fruit?
 Seemingly these items are not considered commonly squeezed and hence it is allowed to squeeze them to sweeten according to all, just as we rule by fruits that are not commonly squeezed.
 However the M”B  rules that one who is stringent to avoid squeezing these types of liquids even into food is to be praised.
 A creamy dip made from fish fat.
 Ketzos Hashulchan 126 Footnote 19
 Shabbos Kehalacha Vol. 2 p. 446
 So rules Tehila Ledavid 320/14
 Shabbos Kehalacha Vol. 2 P. 437
 Based on Admur 302/23
 Shabbos Kehalacha Vol. 2 p. 440
 Shabbos Kehalacha Vol. 2 p. 424
 Shabbos Kehalacha Vol. 2 p. 425
 Shabbos Kehalacha Vol. 2 p. 426