Is refining atk ever worth it?

+3 attack seal I would say, but I have seen a lot of Effies with +Attack in her normal Lance. Idk.

I have seen some QR dragons use it.

I gave my Caeda a +atk wo dao because she needs it, but I'd usually say to use a different refinement and use the seal to patch those ivs

Its usually the worst choice imo,for example,I gave My +spd-HP Olivia a +Atk Wao dao because she actually needed those extra 2 attack to more kill,but it would be ways better instead to have a +Atk Olivia and Upgrade Wao Dao to +spd.I sometimes wonder if I shouldn't have go + spd on my +spd for extra Tankiness.

It's rarely ever good, but it can be better than speed on slow units with high attack (Like Effie) and on extremely fast units with low attack (Like Lon'qu). Generally, attack is valued more than speed because you don't need speed to attack twice in the case of skills like Quick Riposte, Brash Assault, and Breaker skills.

In the case of your Robin, a speed refine is better than an attack refine because he'll want to be hit less more than he wants to hit other hard. If you give him Berkut's Lance with a Speed or Resistance refinement, then he'll become a wall like Gwendolyn, but with slightly better offensive stats at the cost of lower defensive stats. The only downside is that Gwendolyn is easier to get and merge with ideal IV's, and she doesn't really care about speed because everything will deal little damage to her, while letting her charge up her special for damage output.

Conclusion:
If he is in an armored team that uses ward armor on everyone, then he'll be just like Gwendolyn.
However, if he is being used as a standalone unit on a mixed-movement team, you'll want to refine his weapon for more speed and give him an attack+ seal.

In general no however on units that is low on res, and def, it may be better off with attack refine. It also depends on if you want player base boon or def base boon.

As the game progresses, it's usually better off as def boon as less and less units are able to orko. Also def boon tends to be better for TT.