Well, with regard to the specific examples you gave, there are two separate phenomena at play. The first example, "sword" and "lame" is an example of simple linguistic coincidence: two unrelated words happen to sound like one another. The second example, "blanc" and "black," is a result of semantic differentiation from a common root.

Although lame in French does indeed mean "blade," it is not etymologically related to the English word lame: that is, they are not the result of changes in pronunciation to a single common root word. The English word lame comes from a Proto-Indo-European root word, **lem-*, meaning, "to break, broken." The French lame comes from the Latin lamina, meaning a "thin slice of wood, metal, etc.." In English, we use the same Latin root for the word laminate. The source of the Latin word lamina is unknown. Lame and lame are similar merely by the coincidence that the phonetic inventory of human language is limited, and there will inevitably be words that sound very similar to one another because there are only so many combinations of articulations that our physical structures for creating speech can make.

The French word blanc does mean "white," but it is not most closely related to the English black, but rather to the English word blank, which was borrowed directly from Old French. Black and blanc are etymologically distantly related: they both descend from a Proto-Indo-European root *bhel- meaning "to shine, flash, burn." Blanc (and thence blank), as well as bleach, comes from the semantic association of brightness associated with the light given off by fire, while black comes from the semantic association of what something looks like when it has been burned by fire. We see here that two words with entirely different meanings come from a common root, each using a different aspect of that common root word (one, the light coming off of fire; the other, the blackened color of burned objects).

Off the top of my head, I cannot think of any loanwords from French in English or cognates between the two languages that mean opposite things.

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Sword and black come from a different linguistic origin (Germanic) than the French words do (Romance).

French-origin words such as Exercise, jail, mutton, and army among others started coming over into the English language after the Norman Conquest in 1066. Before that, the Aenglisc language (What it was called at the time) was a west germanic language fairly similar to Dutch or Low German, which had been brought over to Britain during the Migration period, when the Angles, Saxons and Jutes took over a large amount of Great Britain, pushing the native languages (Which became Welsh, Cornish, and via migrations of native Britons, Breton) into other areas. Their language thus became the principal language for the area that would come to be known as England (From Englaland, land of the Angles).

From their language, we have words such as Hunt, cow, mouth, ship, house, hand, sun, among many, many others. More words came in from Danish when the vikings began raiding and invading. These words include plunder, berserk, bag, choose and die. Danish (Which at the time was actually Old Norse) also changed some of the grammar, changing some elements of word order (E.G. I will never see you again vs German: Ich werde dich nie wieder sehen).

This is a very simple explanation, but basically some words come from some roots, and despite sounding similar to words in other languages they don't always have similar or even related meanings.

To answer your examples, "Lame" comes from Old English Lama, meaning crippled, while in french it comes from Latin Lamina, meaning a thin piece of metal or wood.

Black comes from Old English Blæc, meaning the same thing; blanc comes from Old French, meaning White. Interestingly enough, this comes from Frankish, another Germanic language. It has the same origin as the word Blank, and in Old English a Blanca was a white horse. It is not, however, connected to the word Black.

Edit: As /u/Dzukian said, Black is extremely distantly related to Blanc.

This answer's based on information from historical linguistics. As is pretty much always the case for questions like this, that means I'll mention 'words' which are not attested historically, but are parts of reconstructed proto-languages. That's what the '*' in front of the old words I'm using indicates.

The English 'lame' comes from a Proto-Germanic term, *lamon, meaning 'weak-limbed', which was itself derived from Proto-Indo-European *lem, meaning 'broken'. There's no etymological connection to French 'lame', it's just a coincidence that they're spelt the same.

There is actually a connection between black and blanc, but it's very far back. Both words are derived, ultimately, from the PIE term *bhleg, meaning "to burn/shine". This then developed in different ways, leading to both of the modern words. Proto-Germanic had *blakaz, meaning 'burnt', from which we got modern English's black. Proto-Germanic also had (from the same PIE root) *blankaz, meaning 'shining, bright', which was borrowed into Latin and from there developed into modern French's blanc, meaning 'white'.