Does the cell have a mechanism to determine DNA sequence from protein?
"Reverse translation"

Translation is the process by which an mRNA is "translated" into a protein.

It is impossible to get the exact DNA sequence from the protein sequence because a large number different DNA strands could code for the exact same proteins. In other words, there is a loss of information in the translation process. See codon redundancy and I am not even talking about introns and UTR.

Now, it is theoretically possible to get one of the large number of possible DNA sequence from the protein sequence. I am not aware of any natural process that does this job.

"Reverse transcription"

Transcription is the process by which DNA is "transcribed" into a mRNA.

It is possible to get the DNA sequence from an mRNA. And it just so happen that such process does exist! A "reverse transcription" if you want. The enzyme that does this job is called reverse transcriptase. Reverse transcriptase is mainly found in retro-viruses but there are a number of non-retro viruses that have a reverse transcriptase.