Who are Nala and Nila in Ramayan?
अयम् सौम्य नलो नाम तनुजो विश्व कर्मणः |
पित्रा दत्त वरः श्रीमान् प्रतिमो विश्व कर्मणः || 6.22.50
(Sea God to Lord Shri Rama) O, excellent man! This one, named Nala, a glorious person, is the son of Vishvakarma; who was given a boon by his father and is equal to Visvakarma.
Nala was the creator of the bridge (Rama Setu or Nala Setu). -Valmiki Ramayana 6.26.21
Nala fought in a extra-ordinary duel with Pratapana. -Valmiki Ramayana 6.43.23
That virtuous Nala swiftly brought water form the sea for Lord Shri Rama's rajatilak. -Valmiki Ramayana 6.128.57
पावकस्य सुतश्श्रीमान् नीलोऽग्निसदृशप्रभ:।
तेजसा यशसा वीर्यादत्यरिच्यत वानरान्।।1.17.12।।
Nila (Neela), the prosperous son of Agni the firegod who equalled him in glow surpassed other monkeys in energy, renown and prowess.
Nila was the Army General of Monkeys. -Valmiki Ramayana 6.3.32/6.59.72
Nila was one of only four great monkeys (Angada, Nila, Sugriva and Hanuman) who could cross the ocean. -Valmiki Ramayana 5.2.30
Prahasta the Army chief of Rakshasas was slain in the battle by Nila. -Valmiki Ramayana 6.59.1
Who are their parents? Are they two brothers and twins? How they got similar kind of name as Nala and Nila?
Their parents from monkeys are unknown. Their fathers from Devas were different Vishvakarma and Agni as described above.
Not sure about twins but they could be brothers (this might be same as how Sugriv (Sun of Surya) and Bali or Vali (son of Indra) were brothers as their father (Riksharaja, who was their mother too) from monkeys was same).
जामवंत बोले दोउ भाई। नल नीलहि सब कथा सुनाई।।
राम प्रताप सुमिरि मन माहीं। करहु सेतु प्रयास कछु नाहीं।।
How did they meet lord Rama?
After Bali's death his youger brother Sugriva became king of monkeys. He later asked all the monkeys to join Rama's army.
How could they make stones floatable on water while building Ram Setu?
The bridge was not built with stones only but trees, rocks and mountains (Source). The floating stones theory doesn't seems 100% true rather it seems the great work of an Engineer/Architect. Nala was son of Vishvakarnma (the great Architect of Devas) and got to know the art from his father. This is what Nala himself says in Valmiki Ramayana:
मम मातुर् वरो दत्तो मन्दरे विश्व कर्मणा |
औरसस् तस्य पुत्रो अहम् सदृशो विश्व कर्मणा || ६-२२-४९
On the mountain of Mandara, the following boon was given by Visvakarma to my mother: "O, god like lady! A son equal to me will be born to you.
औरसस्तस्य पुत्रोऽहम् सदृशो विश्वकर्मणा|
स्मारितोऽस्म्यहमेतेन तत्त्वमाह महोदधिः || ६-२२-५०
न च अपि अहम् अनुक्तो वै प्रब्रूयाम् आत्मनो गुणान् |
I am a son born of Visvakarma's own loins. I am equal to Visvakarma. This god of Ocean has reminded me. The great ocean spoke the truth. Being unasked, I have not told you my details earlier.
समर्थ्श्चाप्यहम् सेतुम् कर्तुम् वै वरुणालये || ६-२२-५१
तस्मादद्यैव बध्नन्तु सेतुम् वानरपुङ्गवाः |
I am capable of constructing a bridge across the ocean. Hence, let the foremost of monkeys build the bridge now itself.
Ramcharitmanas says that the bridge was built by both Nala and Neela:
सैल बिसाल आनि कपि देहीं। कंदुक इव नल नील ते लेहीं।।
बाँधा सेतु नील नल नागर। राम कृपाँ जसु भयउ उजागर।।
But as per Valmiki Ramayana, the bridge was constructed by Nala hence at many places it is referred as Nala Setu:
दशयोजनविस्तीर्णम् शतयोजन मायतम् || 6.22.74
ददृशुर्देवगन्धर्वा नलसेतुम् सुदुष्करम् |
The celestials and Gandharvas, the heavenly musicians saw Nala's bridge, having a width of ten yojanas and a length of hundred yojanas and which was very difficult to be built.