Assalaamu alaykum. I have two questions:
A) We know that singing 'good' songs without instruments is unanimously allowed. So if there is a 'good' song that contains music, can I listen to it in order to learn the song so that I can later sing it without the music?
B) If one grows his beard merely to follow the fashion or in order to follow the fashion as well as follow the Sunnah, is he sinful then?
Please do not refer me to other answers.

All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.

It is not permissible to listen to the Nasheed if it is accompanied by musical instruments, even if listening to it was done for a lawful purpose such as learning the Nasheed, because it is possible to attain this purpose with a means that is free from what is forbidden, such as reading the lyrics of the Nasheed (i.e. in its written form).

Letting the beard grow is not an act of worship that is purely intended for itself in a way that it requires an intention, so there is nothing wrong if someone lets his beard grow to follow fashion or otherwise, but if someone does not intend to get a reward from Allaah through it, then he misses the reward for it. Please, refer to fatwa 318157.

Also, there is no harm if he lets his beard grow with both intentions: following the Sunnah as well as the fashion; the erudite scholar Al-Qaraafi said in Al-Furooq:

“With regard to combining two intentions, such as the one who goes out for Jihaad as obedience to Allaah and to get a share of the booty, then this does not harm him, and it is not forbidden for him by the consensus of the scholars; because Allaah granted him this share from this act of worship, so there is a difference between:

- his Jihaad so that the people would say, 'he is brave,' or in order that the Muslim ruler honors him and gives him abundantly from the Treasury, as this and its like are forbidden Riyaa' (showing off)

- and his Jihaad to get female slaves, horses, and weapons from the wealth of the enemy, as this does not harm him even though he combined the two intentions… yes, these purposes that are associated with the act of worship may decrease the reward, and if the act of worship is pure and not associated with other intentions, then that increases the reward, but there is no sin involved, and the act of worship does not become invalidated...” 

Allaah knows best.