Why is copper(II) sulfate added drop by drop during the Biuret test?
The Biuret Test is a chemical assay that detects the presence of proteins in a sample. The test is not named after any famous scientist, but after an urea dimer called biuret ($\ce{H2NC(O)NHC(O)NH}$). However, funny thing is biuret is not a part of the test at all. The test is called Biuret Test because biuret gives a positive result to the test (See the image below).



The test relies on the characteristic color change (from blue to deep purple or violet; see the image), which confirms the presence of proteins. Biuret isn't a protein, but it gives a positive result to the test, because it has two amide bonds. 

Your question (1): How does $\ce{NaOH}$ dissolve the protein exactly?

Proteins has a lot of amide (peptide) bonds, terminal amine and carboxylic acid groups, and along the $\alpha$-helix, a lot of amino acid side chains including amine groups (e.g., lysine), carboxylc acid groups (e.g., aspartic acid), thiol groups (e.g., cystine), etc. All of them can be reacted with $\ce{NaOH}$ to become ionic, which make it more soluble in water. Specifically, recall most of zwitterionic form of amino acids resist to dissolve in water. Regardless, $\ce{NaOH}$ is essential for the test, not only to make protein more soluble, but also to extract $\ce{H}$ from amide $\ce{N-H}$ bond to make $\ce{N}$ negative (Ref.1).

Your question (2): What is the significance of adding copper(II) sulfate drop by drop?

When you perform the Biuret Test in qualitative manner, you need to be careful to see color change clearly, I'd say picture perfectly. Suppose, you have very dilute protein sample. Adding the blue solution into a colorless solution is tricky, because if you are not slow enough, you won't realize there is a color change happens due to both (before and after) solutions been colored but intensities are low (see the near pipette tip of the picture). If you are doing the quantitative test, it doesn't matter how fast you add the solution. After color development, you use UV-Vis-Spectrometer to measure color intensity: at appropriate wavelength to purple color ($\pu{540 nm}$) where absorbance is zero for the $\ce{CuSO4}$ blue color, thus interference is minimal. 

Your question (3): How would the reaction vary if copper(II) sulfate was added first and $\ce{NaOH}$ added subsequently, drop by drop?

When you search for the Biuret Test, you'd find out that biuret solution is already alkaline with $\ce{NaOH}$ (see the description of solution in the picture). Thus, I'd say, the order does not matter. Yet, if you want to do a crude test with aqueous $\ce{CuSO4}$ solution and $\ce{NaOH}$ solution, you should be careful, because initial $\ce{Cu(OH)2}$ precipitation might happens. If interested, you may read Ref.2 to see the effects of all solutions on the test, which gives very comprehensive review of the test.

References:


H. Sigel, R. B. Martin, “Coordinating properties of the amide bond. Stability and structure of metal ion complexes of peptides and related ligands,” Chem. Rev. 1982, 82(4), 385–426 (DOI: 10.1021/cr00050a003).
A. G. Gornall, C. J. Bardawill, M. M. David, “Determination of Serum Proteins by Means of the Biuret Reaction,” Journal of Biological Chemistry 1949, 177, 751-766 (http://www.jbc.org/content/177/2/751.short).