1a) If you're reading absorbance at 280 nm I don't think urea interferes with your reading. Anyway, you should use the same solution (buffer without protein) for the blank and if you have a standard you should dilute it in the same solution as well. Also, I think I remember that 2 µl drops must be used for proteins assays...

1b) If you're doing a bradford assay, I think you have to dilute your sample to a urea concentration that does not interfere with the bradford reagent (

For some time the Mann group have been using tryptophan fluorescence for peptide and protein measurements. Recently, detailed analysis of the approach was published (Jacek Wisniewski's work):

http://pubs.acs.org/doi/pdf/10.1021/ac504689z

Might be a good idea to try. Solutions with up to 8M Urea could be quantified.

I second point 1a. in Leonardo's reply. However, if you want a dye based assay, which I prefer over nanodrop, then I recommend using BCA instead of Bradford. In my experience its more accurate, robust and impervious to denaturants like Urea, SDS, etc., but it is very sensitive to reducing agents.

Hope it helps, cheers!!

Just a complement of information, you can perform a Bradford assay on a nanodrop, of course! I tried it to measure the concentration of antibodies and it gave me results that were consistent with the reading at 280 nm and with a Bradford assay in a cuvette spectrophotometer...

I will do further steps to remove DNA by 10KDa cut off filtration,in this case can i replace urea buffer by PBS during filtration then measure the protein concentration?

You can use molecular weight cut off columns for buffer exchange but there are issues involved in it. You have to repeat it at least 4 times and each time you will loose around 4 to 5 % of proteins, also sometimes the membranes absorbs the proteins so you might loose more then 20% of proteins. Instead you can try using Biorad's micro bio spin columns. For the estimation of proteins i would suggest you to go for BSA method in micro titre plates. 

I'm not sure about 6M Urea, but Nanodrop should be OK with

There are multiple aspect to your question:

1 OD depends on the extinction coefficient of each protein so if you are measuring a specific protein find its coefficient. Some will have 0 and therefore cannot be measured by OD.

2 If it is a cell extract, you can approximated that 1OD will correspond to 1 mg/ml at 280. In all cases it is an approximation and it is fine if your goal is to load the same amount on a gel since the errors will all be the same.

3 Buffer: I was wondering the same thing myself. I have my proteins in 8M urea. Can I read a mining full OD? The number that I get is much larger than what I would expect. So I measured 15 mg of BSA and resuspended it in 1 ml of my urea buffer. I then did serial dilutions and measured each on the nano drop. There is a BSA setting but you can also use the 1mg/ml setup. The number that I red was very close to the hypothetical value and the 2 fold serial dilution did each increase by 2 fold.

What I suggest you do is do exactly the same with your buffer (Mine does not contain SDS). Make sure that you blank with the extraction buffer that contains all of the protease/phosphatase inhibitor that you use. (In SDS urea you don't need any). The phosphatase inhibit typical absorb a lot at 280.

I know it is a long answer but sometime you have to do the work to get your answer. If you find that you can measure BSA with less than 50% error, it is likely to be enough to use the same amount of protein per well or per digestion.