Learn English – In the phrase “to see how something fares / fairs”, is it “fare” or “fair” that should be used?

First of all, it helps to enclose the phrase in quotes when googling:

• "bear the shame" — 607,000
• "bare the shame" — 83,400

Secondly, having looked through the first 10 pages of the Google results for "bare the shame", exposing would not work in most of those contexts at all, it's quite obviously carrying. And quite a few of those contexts feature extremely poor grammar and punctuation.

Lastly, while the British National Corpus has 1 cite for "bear the shame", and the Corpus of Contemporary American English has 10, neither of them has a single cite for "bare the shame".

So, to me, the answer is pretty clear: "bare" is simply a common misspelling of "bear", whether in the context of this expression or elsewhere.

Dictionaries have long had to contend with this issue. The word run, for example, has 50 or so meanings as a verb, and another 30 or so as a noun, but they all are grouped under one single dictionary entry. On the other hand, bow has three separate entries.

Most print dictionaries denote this using superscripted numerals for each separate entry, much like NOAD does (see screen shot below). In contrast, the online dictionary by Collins uses a numeral in a blue square for each entry to denote the same thing, as can be seen at their listing for bow).

As for your last example:

I came to see the Bishop's see.

I'd say that see and see are not the "same word," based on how they are listed in the dictionary.

However, as John Lawler has mentioned in his comments, it depends on who is counting, and what the count represents. In the sentence:

He had had a cold.

had and had are two separate words (it is a five-word sentence, after all), yet those two hads happen to map to the same dictionary entry, whereas, in the earlier sentence, see and see map to two different dictionary entries.

So it all depends on what your definition of word is.

^{NOTE: Some definitions have been removed from this image in the interest of conserving space}