Can "normal" achievements still be completed with a seasonal character?

With the new seasons in Diablo III, I've started a new seasonal hero, haveing all achievements available for unlocking again.

But I didn't complete all the old achievements yet, so if I kill special enemies for the "Kill al these named enemies" achievements and didn't have them pre seasons yet, does my seasonal kill also count for the normal achievements? Same goes for the "drink from a stupid amount of healing fountains" and enviromental kills,...

Solution 1:

When you achieve something in season mode, you also get the normal mode equivalent achievement.

Just like the example you gave in comments: if you touch 70 healing fountians with a seasonal character while you still needed 50 for the normal achievement, you'll get said achievement both in season and normal modes.

Also, from the wiki:

Will achievements earned in season mode also be rolled over to non-season and granted there (if you didnt previously have them in non-season)?
Under current design, if you unlock an achievement on a Seasonal hero that you haven't already unlocked on a non-Seasonal hero, you'll gain credit for both at the same time.

Solution 2:

At the end of the season, any achievements you earn which were not earned in non season will be awarded

Your heroes keep all their gear, crafting materials, gold, blood shards, Paragon experience, and any new recipes they may have earned during the course of a Season. This also includes any new Stash tabs purchased or Artisan levels earned (if higher than your non-Seasonal profile). Finally, if you've managed to earn achievements during a Season that you've never conquered before, you'll earn them in the corresponding non-Seasonal mode.

http://us.battle.net/d3/en/blog/15198713/