I was wondering if either of these were actually doable in DA:O.

1. Successfully completing both Bhelen's and Harrowmont's 1st tasks.

2. If a warden mage can somehow become 1st Enchanter. I thought no but I've heard Wynne has a conversation with a mage character where she may encourage him/her to become 1st Enchanter due to a combination of youth (will be around for a while) and greater than typical experience (your warden has traveled much more & dealt with more people than many mages.

I thought I was able to do several of Bhelen and Harrowmont's tasks, but it has been a while, so maybe I am mistaken and I only got the quests and didn't finish them.

1. You can (technically) finish their first tasks, but doing so will lose you the favor of both factions. If you don't want to lose the support of the dwarves, you should not speak to Lord Dace in the Deep Roads (you can still visit and clear the area he's in). There is no equivalent for Harrowmont, iirc.
What I did with my warden was that I ultimately triple-crossed Bhelen - first by going to Harrowmont after speaking with Bhelen's agent, second by supporting Harrowmont even after agreeing to act as a double agent for Bhelen (you'll also get some interesting quests at this point) and finally after deciding to give Harrowmont the crown. There's actually a complex but rewarding chain of quests you can follow if you decide to play both factions - just don't speak with Lord Dace, and don't forget to visit Bhelen's agent after you finish Harrowmont's first task (and every time Harrowmont's agent gives you a quest).

2. Between the Blight, Grey Warden concerns and other darkspawn threats, the Warden simply doesn't have the time.

Thanks for the attempted answer.

I tried to do both 1st tasks and couldn't seem to make it work.

My preference is to double cross Bhelen for the 2nd task (act as Harrowmont's double agent) and then give Bhelen the crown.

However I would like to complete Harrowmont's 1st task prior to doing this.