Do either Arjuna or Bhima ever do pure acts of kindness or charity?
Not counting killing bad guys to help people, like the Bakasura case.
The way Arjuna acted towards Ekalavya was inexcusable.
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There was a rule, Pandavas made in presence of Narada:
the rule they made was that when one of them would be sitting with Draupadi, any of the other four who would see that one thus must retire into the forest for twelve years
As per Mahabharata: Adi Parva: Arjuna-vanavasa Parva, a Brahmana asked help from Arjuna in rescuing his cows from thieves. At that time Arjuna's weapons were in Draupadi's chamber where Yudhisthira was there with her. Arjuna went there to get the weapons to help the Brahamana even though he knew that doing that will result into 12 years exile.
"After some time, O king, it so happened that certain robbers lifted the cattle of a Brahmana, and while they were carrying away the booty, the Brahmana, deprived of his senses by anger, repaired to Khandavaprastha, and began to reprove the Pandavas in accents of woe.
The Brahmana said, 'Ye Pandavas, from this your dominion, my kine are even now being taken away by force by despicable and wicked wretches! Pursue ye the thieves. Alas, the sacrificial butter of a peaceful Brahmana is being taken away by crows! Alas, the wretched jackal invadeth the empty cave of a lion! A king that taketh the sixth part of the produce of the land without protecting the subject, hath been called by the wise to be the most sinful person in the whole world. The wealth of a Brahmana is being taken away by robbers! Virtue itself is sustaining a diminution! Take me up by the hand, ye Pandavas for I am plunged in grief!"
"Vaisampayana continued, 'Dhananjaya, the son of Kunti, heard those accents of the Brahmana weeping in bitter grief. As soon as he heard those accents, he loudly assured the Brahmana, saying, 'No fear!'
But it so happened that the chamber where the illustrious Pandavas had their weapons was then occupied by Yudhishthira the just with Krishna. Arjuna, therefore, was incapable of entering it or, going alone with the Brahmana, though repeatedly urged (to do either) by the weeping accents of the Brahmana. Summoned by the Brahmana, Arjuna reflected, with a sorrowful heart, Alas, this innocent Brahmana's wealth is being robbed! I should certainly dry up his tears. He hath come to our gate, and is weeping even now. If I do not protect him, the king will be touched with sin in consequence of my indifference; our own irreligiousness will be cited throughout the kingdom, and we shall incur a great sin.
If, disregarding the king, I enter the chamber, without doubt I shall be behaving untruthfully towards the monarch without a foe. By entering the chamber, again, I incur the penalty of an exile in the woods. But I must overlook everything. I care not if I have to incur sin by disregarding the king. I care not if I have to go to the woods and die there. Virtue is superior to the body and lasteth after the body hath perished!' Dhananjaya, arriving at this resolution, entered the chamber and talked with Yudhishthira.
Coming out with the bow, he cheerfully told the Brahmana, 'Proceed, O Brahmana, with haste, so that those wretched robbers may not go much ahead of us. I shall accompany thee and restore unto thee thy wealth that hath fallen into the hands of the thieves.' Then Dhananjaya, capable of using both his arms with equal skill, armed with the bow and cased in mail and riding in his war-chariot decked with a standard, pursued the thieves, and piercing them with his arrows, compelled them to give up the booty.
Benefiting the Brahmana thus by making over to him his kine, and winning great renown, the hero returned to the capital.
Bowing unto all the elders, and congratulated by everybody, Partha at last approached Yudhishthira, and addressing him, said, 'Give me leave, O lord, to observe the vow I took. In beholding thee sitting with Draupadi, I have violated the rule established by ourselves. I shall therefore go into the woods, for this is even our understanding.'
So, this was one of the pure acts of kindness or charity for which Arjuna had to spend 12 years in exile.