Transferring points acquired by unlocking achievements to Bioware Points

The best source I have for this is the Fallout Wikia. They say:

Please note that cheating, on the PC version at least, appears to disable achievements for that play. If you are cheating (or using console commands to get around bugs) , you won't get any achievements unless you save, quit, reload, and redo the achievement.

Unfortunately I can't test this myself, but this seems to stem from experimentation.

Oh, seeing as you mentioned it: I'm fairly sure using cheats will not remove any achievements you've earned before. That's just the way Steam works.

Update: Using mods does not affect achievements at all. Only console commands are construed as cheating.

Console commands do not affect achievements in Skyrim. I've used it to clip through places when I ended up falling and becoming stuck and still received achievements in the same session. I've also togged tgm on occasion while testing stuff and still received achievements after I finished testing without restarting my game.

Once you get past all the speculation, this thread seems to confirm that using console commands do not affect achievements at all.

Of note from that thread:

• One person got the buy a house achievement right after giving himself 5000 gold via additem.
• Another person still continued to receive achievements despite walking around with an max encumbrance weight of 5000 via modav.