Is it permitted to make wudu during adhan?
Yes, as this is Mubah and there is no proof in the Shariah that prohibits this.
Is it permitted to respond to adhan while making ablution?
Yes, as to repeat the Adhan should be by anyone in whatever state they are as the Adhan is Zikr, the exception is those who are in Salat, or in the toilet, or during the fulfilment of the desire with one mate.
The Prophet ﷺ said:
إذا سمعتم النداء، فقولوا كما يقول المؤذن
When you hear the Mu'adhdhin (pronouncing the call to Salat), repeat
after him what he says.
Also Annawawi said in المجموع شرح المهذب:
قال أصحابنا : ويستحب متابعته لكل سامع من طاهر ومحدث وجنب وحائض وكبير
وصغير ; لأنه ذكر . وكل هؤلاء من أهل الذكر ، ويستثنى من هذا المصلي ومن
هو على الخلاء والجماع
NOTE: the following is not exact translation:
Our companions said: and it is liked to repeat it for whoever hears it
from one who is pure to one who is impure, from old to young; because
it is Zikr. and all of the mentions are of the people of zikr, and
the exception are those who are making salat and those who are in the
toilet or with his wife.