Minecraft – On Minecraft, is it possible to breed the wolf with the friend’s wolf? they’re both already tamed

Yes, but you have to have exactly the same mods, with all the exact same version numbers and configuration, except for some mods that are client-side only (e.g. minimaps, Optifine, MAtmos, etc.) which don't need to be the same or even present on both sides.

The best way to test this isn't to check every mod you have — the easiest way is to just try. Assuming you're using Forge to install Minecraft mods, any mismatch with the hosting player's Minecraft mods will be detected by Forge, and it will refuse their connection with a message telling them what they're missing or need to get rid of before trying to connect again.

Short answer: No, you don't need a Farmer. But, for two villagers, you need at least 6 doors (that lead to a covered area).

Long answer:

The key to finding the answer is found within your own question. "Additionally, ..." implies more information came before your quoted part of the Wiki.

From that same linked wiki: "Villagers will mate depending on the number of valid doors" Make sure you have enough. You need about 3 times more doors than villagers for them to even consider breeding. This was the sole factor in breeding in previous versions. Now, it's only the first part.

Also, my intuition says the chunk that contains the new village needs to be loaded for breeding to occur. This means you need to be close to the villagers for longer periods of time.

Lastly, they need to be close to each other to breed.

Hope that helps!