Probably, up to a point.
Firstly, I have to start with a point of order from Snape:
"Only Muggles talk of 'mind-reading'. The mind is not a book, to be opened at will and examined at leisure."
(Order of the Phoenix, Chapter 24, Occlumency).
It's important to clarify this: neither Harry nor Nagini could read Voldemort's mind. That's because mind-reading isn't really a concept in Harry Potter. The closest concept is Legilimency, which wizards (not snakes) could theoretically employ against Voldemort, although he was such a powerful wizard that this would likely be a fruitless exercise.
However, Harry and Nagini are in a unique category since
they both contain a piece of Voldemort's soul.
This means that Nagini may enjoy access to Voldemort's thoughts in a way that's similar to the way Harry does, but I don't think we can be 100% sure since the Nagini-Voldemort thought relationship isn't given nearly as much attention as the Harry-Voldemort one. So the question is, what access did Harry have to Voldemort's thoughts and did Nagini have the same access?
Harry had momentary insights into Voldemort's thoughts. These occurred:
- When Voldemort's emotions were running high. Harry could see where Voldemort was and feel his anger, joy, frustration etc.
- When Harry/Voldemort were asleep. Harry had insights into Voldemort's dreams during Order of the Phoenix when Voldemort was dwelling on the Department of Mysteries. He also had two dreams about Voldemort during Goblet of Fire.
- When Voldemort planted false thoughts into Harry's mind at the end of The Order of the Phoenix.
- When Harry chose to dip into them. There's a huge caveat to this one in that this was an ability that Harry only gained right at the end of Deathly Hallows. He could also only choose to access Voldemort's thoughts when Voldemort was emotional (as per point 1).
Harry closed his eyes, and as his scar throbbed he chose to sink again into Voldemort's mind.
(Deathly Hallows, Chapter 30, The Sacking of Severus Snape).
Note that these examples overwhelmingly show Harry as the passive party. He can't access Voldemort's thoughts at will. He just gets glimpses when the conditions are right. I think that we can tentatively say that Nagini enjoyed a similar relationship with Voldemort. She didn't have an unbroken connection to Voldemort's every thought. But it's likely that she could, say, feel Voldemort's anger brewing just as Harry could. She would probably have sensed his euphoria when the Azkaban breakout happened. And so on.
What we can say with confidence is that she had a telepathic connection with Voldemort that allowed her to communicate with him non-verbally, even when they were separated by great distances. She uses this at Godric's Hollow to tell Voldemort that she has Harry, without using a Dark Mark or saying anything out-loud.
Then she closed her eyes and several things happened at once: Harry's scar prickled painfully, the Horcrux twitched so that the front of his sweater actually moved; the dark, fetid room dissolved momentarily. He felt a leap of joy and spoke in a high, cold voice: hold him!
(Deathly Hallows, Chapter 17, Bathilda's Secret).
This isn't mind-reading but it is certainly evidence of a strong mental connection between the two. That's really all that we can definitively say that they had.
As for the other question about Nagini using their connection to protect Voldemort: no, Nagini never protected Voldemort. (Except indirectly by protecting him from death in her role as a Horcrux, that is).