Emotional Support without God in Theravada In religions with God, the believer or devotee can call on God for help during distress, pray to God to seek emotional relief, find a virtual shoulder to cry on and sing God's praises or glories for emotional upliftment (including the use of devotional music).

They also establish a personal relationship with God. In Christianity, one has the relationship of being a child of God. In Islam, one has the relationship of being a servant of God, created by Him and therefore must submit to Him. In Hinduism (Hare Krsna), one can see God as a friend, son, teacher, master, lover etc.

Whether God exists or not is not the point here for discussion. Certainly in the above cases, there may not be any tangible direct interaction with God, but the believer or devotee could often indirectly feel the presence of God or see good things that happen to them as miracles that come as a response to devotion and prayer.

The above describes the emotional support that one could get from religions that have a God. I know this exists in certain parts of Mahayana and Vajrayana Buddhism too with Kwan Yin (Bodhisattva Avalokiteshvara) and other Buddhist deities fulfilling the emotional support role of God. Chinese Buddhists have long composed songs and prayers to sing the glories of Kwan Yin, and they also call on Her during distress.

However, in Theravada Buddhism, there exists no such emotional support from any deity. So, what should the Theravada Buddhist do in these cases? For example, during sudden distress or grief or loneliness, how would the Theravada Buddhist seek emotional support or relief, without a deity? How does a Theravada Buddhist sooth his or her emotions without devotional music? With whom does the Theravada Buddhist seek a relationship when they find that their worldly relationships have failed them?
You point out several supposed benefits of God and then ask how the godless fare without these benefits. I would first challenge the virtue of the benefits mentioned:




  In religions with God, the believer or devotee can call on God for help during distress


Unless you believe God directly intervenes, for which there is no evidence whatsoever, calling on God for help seems as useful as calling on a milk carton for help. This cannot be accurately seen as a benefit, since it most obviously sets one up for disappointment and unreasonable expectation.


  He who has eyes can see the sickening sight;
  Why does not Brahma set his creatures right?
  
  If his wide power no limits can restrain,
  Why is his hand so rarely spread to bless?
  
  Why are his creatures all condemned to pain?
  Why does he not to all give happiness?
  
  Why do fraud, lies, and ignorance prevail?
  Why triumphs falsehood,—truth and justice fail?
  
  I count your Brahma one th’ injust among,
  Who made a world in which to shelter wrong.
  
  -- Jat. 543 (Cowell, trans)





  pray to God to seek emotional relief


Prayer is a lot like meditation so it can certainly have beneficial side effects, but since it relies on the presumption that an imaginary being will prevent harm or bring fortune, it falls under the same pitfalls as the first benefit. It is a lot like depending on a father figure to protect one - it is only beneficial if one's father actually exists.




  find a virtual shoulder to cry on 


By virtual, we would understand you to mean "imaginary". There is some benefit, surely, in imaginary friends, but in the end we would see this as a dependence. Catharsis is not recognized as beneficial in Theravada Buddhism; it is habit forming, and thus crying on anyone's shoulder, real or imaginary, is unwholesome behaviour that leads to craving, clinging, and further becoming.




  and sing God's praises or glories for emotional upliftment (including the use of devotional music).


We do something similar in Buddhism, praising the virtues of the Buddha, and our chanting is similar in nature to this. So, this I would recognize as a virtue, qualified by the fact that the virtues of God are not all appealing to a Buddhist - believing God to be all-powerful, for example, is a sign of delusion, and not at all wholesome.

Either way, there is a similar practice:


  I also say unto you O monks — if any fear, terror or hair standing on end should arise in you when you have gone to the forest or to the foot of a tree, or to an empty house (lonely place), then think only of me thus:
  
  "'Such Indeed is the Blessed One, arahant (Consummate One), supremely enlightened, endowed with knowledge and virtue, welcome being, knower of worlds, the peerless trainer of persons, teacher of gods and men, the Buddha, the Blessed One.' Monks, if you think of me, any fear, terror, or standing of hair on end, that may arise in you, will pass away.
  
  -- SN 11.3 (Piyadassi, trans)





  They also establish a personal relationship with God. 


Again the problem of an imaginary friend. Buddhism doesn't rely upon even the Buddha for support; as above, it's not that having imaginary friends is bad, it's that reliance on others is impractical, since all things change, including God, Buddha, etc.


  And those monks who had not yet overcome their passions wept and tore their hair, raising their arms, throwing themselves down and twisting and turning, crying: ‘All too soon the Blessed Lord has passed away, all too soon the Well-Farer has passed away, all too soon the Eye of the World has disappeared!’ But those monks who were free from craving endured mindfully and clearly aware, saying: ‘All compounded things are impermanent — what is the use of this?’
  
  Then the Venerable Anuruddha said: ‘Friends, enough of your weeping and wailing! Has not the Lord already told you that all things that are pleasant and delightful are changeable, subject to separation and to becoming other? So why all this, friends? Whatever is born, become, compounded is subject to decay, it cannot be that it does not decay. The devas, friends, are grumbling.’
  
  -- DN 16 (Walshe, trans)





  Certainly in the above cases, there may not be any tangible direct interaction with God, but the believer or devotee could often indirectly feel the presence of God or see good things that happen to them as miracles that come as a response to devotion and prayer.


Feeling the presence of God is just a feeling; one's association of one's feelings with the existence of God is merely conjecture. Buddhists have the same sort of feelings, but they recognize them simply as feelings. It is not considered a positive activity to cling to such feelings or ascribe meaning to them beyond their existential nature, since that leads to craving, clinging, and further becoming.

Seeing the good things that happen to one as miracles is pure speculation; it is often accompanied by (similarly speculative) rationalization when bad things happen. Neither is particularly wholesome, since they lead to unreasonable expectations and disappointment. More directly, they ignore the fact that there are perfectly natural causes and conditions that lead to all of one's experiences, and thus fail to attain understanding about the nature of reality - chalking it all up to "God's handiwork". This is a bad thing, since an understanding of reality is necessary to free oneself from suffering.


  ‘This, monks, the Tathāgata understands: These viewpoints thus grasped and adhered to will lead to such-and-such destinations in another world. This the Tathāgata knows, and more, but he is not attached to that knowledge. And being thus unattached he has experienced for himself perfect peace, and having truly understood the arising and passing away of feelings, their attraction and peril and the deliverance from them, the Tathāgata is liberated without remainder.
  
  -- DN 1 (Walshe, trans)




So, your supposed benefits are, in all, not very impressive to a Theravada Buddhist. As to your questions:


  For example, during sudden distress or grief or loneliness, how would the Theravada Buddhist seek emotional support or relief, without a deity? 


We don't seek relief from emotions, we seek to understand emotions. It is by understanding something that one becomes free from it. The need for relief implies aversion, which becomes habit forming and leads to craving, clinging, and further becoming.


  How does a Theravada Buddhist sooth his or her emotions without devotional music? 


Same. Also, devotional music leads to sensual attachment, thus - you guessed it - craving, clinging, and further becoming.


  With whom does the Theravada Buddhist seek a relationship when they find that their worldly relationships have failed them?


There is an underlying assumption here in the importance of relationships. In fact, Theravada Buddhism takes relationships quite seriously. As acknowledged, having a relationship with an imaginary friend can be helpful - a Buddhist's relationship with the Buddha is similar to this. But Theravada Buddhism sees more importance in one's relationship with one's teacher who is a tangible source of knowledge, advice, and exemplary behaviour very much unlike one's mental construction of God or the Buddha.


  "Know I am Nārada by name,—a kassapa; my heavenly rest
  I have just left to tell thee this,—to associate with the wise is best.
  
  -- Jat. 539 (Cowell, trans)