Answer: Please find answers in explanation column.
Explanation:
Our Causiana tree is a poem of 5 Stanzas and 55 lines by Toru Dutt who described  her childhood memories in India centering about a tree, The Casuarina tree which she expressed  her love for it .
What are the birds that sing throughout the day?
---- The birds that sing all day are the Kokilas as we can see from Stanza two , Line 7 which is 
"And far and near kokilas hail the day;" 
Kokila is a type of Asian cuckoo bird called the  Asian Koel  found  mostly in  India , China, and Southeast of Asia. 
How do we know the water-lilies are white?
We know the water lilies are white because The writer compared the  water-lilies by the hoar tree to gathered snow. and we know that Snow is white in colour.  This we can see also in Stanza 2, but line 11.
"The water-lilies spring, like snow enmassed."