Did Gandalf think that Frodo was dead?

No

Gandalf knew Frodo was carrying the Ring. Regardless of what the Mouth of Sauron tried to bully them with, Gandalf knew that as long as this was happening, Frodo was in Mordor with the Ring and Sauron still thought it was with the host of the West. (Gandalf also would've felt the pull of the One Ring if Sauron had worn it again, kindly reminded by Ian Bush). In addition, the Mouth of Sauron reveals he has far less information than he believes:

‘So you have yet another of these imps with you!’ [...] it is plain that this brat at least has seen these tokens before, and it would be vain for you to deny them now.’
Indeed, I know them all and all their history, and despite your scorn, foul Mouth of Sauron, you cannot say as much.
_{Return of the King, Book V, Chapter 10: The Black Gate Opens}

Gandalf shows that he has more information and knowledge of Frodo's belonging than the Mouth of Sauron, as he is aware there is one crucial piece of Frodo's belongings missing, that which is most valuable to Sauron. Gandalf did not have anything to fear as as long as they thought Frodo was a spy, Gandalf knew he was indeed still on his errand.

Further evidence to the Mouth of Sauron’s cluelessness is shown in his lack of knowledge of Frodo’s errand:

Or else his errand was one that you did not wish to fail? It has.
_ibid.

This would again be clear to Gandalf that the negotiator doesn’t hold all the information, the suggestion being that Frodo’s spying has failed, unaware that Frodo’s errand is the destruction of the Ring. Again Gandalf has no need to fear any harm has come to Frodo.

With regards to the emotion described for the Host of West there are two important instances, the first:

No-one answered him, but he saw their faces grey with fear and the horror in their eyes, and he laughed again, for it seemed to him that this sport went well
_ibid.

The first key point here is that “it seemed to him” [The Mouth of Sauron] that he’d succeeded, but as shown above he was gravely mistaken due to a lack of information. Secondly this passage doesn’t suggest that Gandalf’s face turned “grey with fear” but describes the collective group that had ridden up with him.

A later quotation suggest Gandalf’s specific reaction of defeat, however this may just be a trick of Gandalf’s to truly get his will in the conversation and find out about Frodo’s whereabouts. Or could be an issue of unreliable witnesses as we had seen at Orthanc.

Gandalf finally, asks the Mouth to produce Frodo, after hesitation Gandalf ends the parlay knowing not only does Sauron not have the Ring, but he also doesn’t have Frodo. Throughout the conversation, Gandalf continually proves for himself that he has little to fear as Frodo has not been captured by Sauron, the Ring is still with the Free People's and in addition, the Mouth of Sauron seems to have no knowledge of the existence of Samwise the Great.

‘These we will take in memory of our friend,’ he cried. ‘But as for your terms, we reject them utterly. Get you gone, for your embassy is over and death is near to you.
_ibid.