This is an incredibly vague question. If you're running a gel (be it agarose or polyacrylamide), not really because the EtBr is in so much excess. In this case, the only difficulty is for the EtBr to intercalate between the bases and then fluorescence should be possible - there aren't many scenarios where methylation impairs this (methylation doesn't usually interrupt the pi stacking that occurs between the conjugated ring structures of the bases and EtBr). Now, if you're asking about the process of DNA methylation, there is some work to suggest that the addition of EtBr may interfere with methylation. So, if you're looking at a gel, you're golden. If you're working on epigenetics, probably not so much.

I have no idea what you're asking about with single strand plasmid synthesis in relation to EtBr. It is likely that single strand synthesis is still possible once EtBr is bound to a plasmid's dsDNA but there will probably be an increased mutation rate. If you clarify this question, I bet the friendly fellows of researchgate will be more than willing to comment.

according to me, methylation should not affect binding of ETBR because bacteria DNA is methylated but still ETBR binds to bacterial DNA.  and ETBR  intercalate between two bases in the hydrogen bonds so..i think methylation it does not affect ETBR binding...  you can read more information in "Molecular Biology of Gene by Watson" book...

No problem: DNA methylation does not affect EtBr binding. Reformulate the second question.

I completely concur with Patrick on the answer that DNA meth will NOT interfere with EtBr binding in vitro, when in excess. In vivo there might be other effects that are caused due to compaction issues of heterochromatin, and hence the answer is complicated depending on the epigenetic status of the area of chromatin being studied.

Lastly. your Q about single stranded plasmid synthesis, yes - if you look up M13 based vectors - they are meant precisely for this reason. Depending on whether you use a PLUS strand primer or minus strand primer - you can get synthesis of a single strand from the vector in vitro. In vivo - depending on the orientation of the strand, one will get synthesized when aided with M13 phage replication machinery. (look up basic protocol books such as Molecular Cloning by Sambrook and Maniatis, for detailed protocols and concepts)