Word Usage – Understanding the Word ‘Unlockable’

If you check the definitions of distinguish, the meaning varies depending on the statement.

In the case of:

1) to separate or mark off by differences; perceive or show the difference in; differentiate

When you write:

I can't distinguish the mountain you climbed in this picture.

I expect to read "distinguish between (something) and (something else)".

I can't distinguish between the mountain you climbed in this picture and the one I visited last year.

In the case of:

3) to perceive clearly; recognize plainly by any of the senses

When you write:

This tree distinguishes the mountain we climbed in this picture.

if the tree was iconic so that when you saw it you knew where it was, then this usage is OK.

I guess this one applies to your example, without additional context.

Your understanding of identify is correct.

Words such as fudging, freaking, fricking, and flipping are euphemisms for fucking. Here's an entry on "flip" (my emphasis):

flip (v.)
1590s "to fillip, to toss with the thumb," imitative, or perhaps a thinned form of flap, or else a contraction of fillip (q.v.), which also is held to be imitative. Meaning "toss as though with the thumb" is from 1610s. Meaning "to flip a coin" (to decide something) is by 1879. Sense of "get excited" is first recorded 1950; flip (one's) lid "lose one's head, go wild" is from 1949, American English; variant flip (one's) wig attested by 1952, but the image turns up earlier in popular record reviews ["Talking Boogie. Not quite as wig-flipping as reverse side--but a wig-flipper" Billboard, Sept. 17, 1949]. Related: Flipped. Flipping (adj.) as euphemism for fucking is British slang first recorded 1911 in D.H. Lawrence. Flip side (of a gramophone record) is by 1949.
(Etymonline)

It's not exclusive to BrE, as it's also heard in AmE. However, it might be less common.

They are used just like you said (for emphasis) and they're used when offensive language is not allowed or not called for. They seem like swear words because they are, just milder ones.

It has nothing to do with "flipping someone off."

As the the entry above suggests, "flipping someone off" likely comes in part from the fact that flip/flipping can be used to describe the movement of extending a finger in that manner.