Learn English – What’s the idiom for getting lost in a (malfunctioning) bureaucracy

I believe that strategy would generally be referred to as "delaying tactics".

Some comments mentioned "running out the clock", which is a sports metaphor from timed sports. I've heard it used as a metaphor in other arenas, although its most applicable when you know exactly how long you need to delay.

In the work arena specifically, there is a tactic called work-to-rule, where the employee gums up the works by following every single workplace rule to the letter.

I don't think we're going to find a verb to specifically mean sing a song of grief (although I hope someone discovers one).

However, there are two nouns - elegy and dirge - which describe such a song. And there is the verb lament which is broader than just singing, but would include singing.

to sing an elegy - I think the best option to describe the form of music in your example

a sad poem or song : a poem or song that expresses sorrow for someone who is dead

http://www.merriam-webster.com/dictionary/elegy

to sing a dirge - could work, but often implies funeral

a song or hymn of grief or lamentation; especially : one intended to accompany funeral or memorial rites

http://www.merriam-webster.com/dictionary/dirge

lament - if you want to describe the way in which someone is singing in your example, then I think this would fit

Express passionate grief about

http://www.merriam-webster.com/dictionary/lament