What boons did Ravana receive from Brahma?
Ravana was advised by his mother Kaikasi to do penance to become equal in to his brother Kubera. Then thee brothers Ravana, Kumbhakarna and Vibhishana started doing penance. After some time lord Brahma appeared before them. Ravana received the boon that he will not be killed by Deva, Gandharva, Naga, Yaksha, Danavas. This has been described in Uttara Kanda of Ramayana, Sarga 10.
Thus addressed by the creator, Brahma, the Ten necked one, O Rama, standing before him with joined hands, said, O lord of creatures, I would, O eternal one, be incapable of being slain by Suparnas and Nagas, Yakshas, Daityas, Danavas and Rakshasas, and the Devas ;for, O thou that art worshipped by the immortals, anxiety I have none from any other beings. Indeed, I deem as straw creatures such as men etc Thus accosted by the Raksha the Ten necked one that righteous souled one, the Great father, along with the Devas, said, O foremost of Rakshasas, what thou sayest shall come to pass.
Having, O Rama, said this unto the Ten necked one, the Great father (again spoke), Hear !I, having been gratified, will confer on thee a fresh boon. O Rakshasa, O sinless one, those heads of thine which have been offered as sacrifices and which have sunk into the fire, shall again be thine..
And, O placid one, I shall also confer on thee another boon difficult of being obtained, The form that thou shalt wish to wear, shall instantly be thine.
Same thing has been discussed in Ramayana, Bala Kanda, Sarga 15.
तेन गन्धर्व यक्षाणाम् देवतानाम् च रक्षसाम् |
अवध्योऽस्मि इति वागुक्ता तथेति उक्तम् च तन् मया || १-१५-१३
"Ravana said while seeking boon that, 'I shall not be killed by gandharva-s, yaksha-s, or by gods, or by other demons...' and I also said 'so be it..." [1-15-13]
Which boon gave him mystical powers and strength ?
Ravana has his mystical and physical strength from birth because he had born from the womb of a Rakshasi (Kaikasi). This is described in Ramayana, Uttara Kanda, Sarga 9.
In the meanwhile.O Rama,that twice born one Pulastya s son was performing the Agnihotra, like the fourth Fire itself. And without minding that terrific time, Kaikasi(), having regard to the dignity of her sire, coming up before him, stood (there) hanging her head down towards his feet and throwing up the earth with her great foe.
And seeing that one of shapely hips, having a face fair as the full Moon, (that exceedingly high minded ascetic) flaming in energy accosted her thus, O gentle on, whose daughter art thou ?And whence dost thou come hither ?And what is thy errand ?And for whom(dost thou come) ?O beauteous(damsel), truly tell me this. Thus addressed, the girl,with joined hands, said, O ascetic, thou art competent to get at my intent by virtue of thy own power. Yet, O Brahmarshi, know me as having come here at the mandate of my sire.
My name is Kaikasi. The rest do thou read thyself. And thereupon, the ascetic, entering into contemplation, said these words, O gentle lady, I have learnt the purpose that is in thy heart.
O thou having the gait of a mad elephant, there reigns a powerful desire in thee for having ofspring. Inasmuch as thou hast come to me at this fierce hour, hearken, thou amiable one, as to the kind of ofspring that thou shalt bring forth. Thou shalt, O thou of graceful hips, bring forth terrible and grim visaged Rakshasas delighting in frightful friends, and of cruel deeds.
Hearing his speech, she, bowing down, said, O reverend (ascetic), such sons of terrific ways seek I not from thee that followest the Veda. Therefore it behoveth thee to favor me. On being thus besought by the girl, Visrava, best of ascetics, again addressed Kaikasi, like the full Moon addressing Rohini, O fair faced one, the son that thou bringest forth last, shall be like unto my line, he shall, without doubt be righteous souled.
Having been thus accosted, the girl, O Rama, after a length of time brought forth a very terrible and hideous ofspring having the form of a Raksha, having ten necks, furnished with large teeth, and resembling a heap of collyrium, with coppery lips, twenty arms, huge faces, and flaming hair. On his having been born, jackals with flaming mouths and other ferocious beasts began to gyrate on the left. And that god showered down blood ;and the clouds uttered forth harsh sounds.
And the Sun was deprived of his splendour ;and meteors began to dart to the earth. And the earth shook ;and the wind swept away violently. And that lord of streams the ocean, which was calm before, became agitated.
And his sire resembling his grand father named him,(saying), As this one hath been born with ten necks,he shall be called Ten necked.
Did he receive any boons from Goddess Durga too?
There is no mention of Goddess Durga giving boon to Ravana in Valmiki Ramayana. May be some Puranas talk about this.
