Question about core cards

That depends on your long-term plans. If you only want to play in Standard and not Wild or Classic, then sure, it's free dust. However, you won't keep Core set cards if/when they leave the Core set in future rotations, so you'll need your own copies (e.g. from the Classic set) if you want to continue using them in Wild and Classic.

So from what I've seen, most people who play Wild and Classic are keeping their copies so they don't lose the cards later down the line.

Obviously Battlegrounds is completely unaffected, and Duels wouldn't lose very much.

Thanks.

I personally recommend against disenchanting any classic cards purely coz it roadblocks you from playing certain play modes like classic/wild/duels/tavern brawl/future game modes. Also remember that tavern brawls will reward a Year of the Phoenix pack so obtaining free classic packs will be non-existent now. 

If you're super tight on dust, another recommendation is looking at HSReplay or here on OoC to see the best performing classic cards and keep those while disenchanting the rest.