Parts B and C have errors. In cases of syntax like 'neither..nor', the verb must be in line with the nearest noun, but the auxiliary verb 'was' is singular which follows the plural noun 'eggs'. So, 'were' should be used instead. In part C, the verb 'open' is in the simple present tense, but the sentence is in the past tense ('was' is a past tense verb), so a past tense form of the verb 'opened' has to be used.
The correct sentence is: Neither the chicken nor its eggs were seen when I opened the cage's gate.
Hence, the correct answer is option (d).