The urea will speed up decomposition provided that other elements are in sufficient supply. Yes. I expect the N to be largely converted to microbial protein unless you apply excess. The end point for a stable material will likely be ~12 C: 1 N.

Yes you would expect some initial N draw down to assist with decomposition of high C:N ratio sawdust (and as Paul has suggested to bring C:N ratio close to 12). As a general rule of thumb (but there are plenty of 'what ifs'), you might expect 80% of a low C:N ratio plant residue like grass clipping to be decomposed (C released) within 12 month, where maybe 50% of a waste derived compost with a higher C:N ratio would be decomposed in the first 12 months. The immobilisation and subsequent rate of mineralisation will follow a similar trend depending on how much N you add. So how you manage your soil inputs of C (sawdust) and N (as urea) depends on what you want to achieve (prevent N leaching, maximise sawdust decomposition, minimise sawdust decomposition, improve organic matter content and soil physical properties, maximise plant productivity) which you didn't mention.

Please see this paper.

Waste Manag. 2004;24(8):805-13. Effect of C/N on composting of pig manure with sawdust. Huang GF1, Wong JW, Wu QT, Nagar BB. Author information 1 College of Natural Resources and Environment, South China Agricultural University, PR China. Abstract The aim of this composting trial was to evaluate the effect of C/N on the composting process of pig manure with the purpose of reducing the amount of sawdust normally used as co-composting materials. Two aerobic static piles were prepared consisting of pig manure mixed with sawdust at an initial C/N of 30 (pile A) and 15 (pile B), respectively. Pile B containing larger amount of pig manure showed a slower rise in temperature, lower maximum temperature, and shorter thermophilic phase than pile A. It also resulted in higher pH and electrical conductivity (EC) values, and even higher contents of soluble NH4-N and volatile solids throughout the composting period. Chemical and biological parameters including dissolved organic carbon (DOC) (4932 mg kg(-1)), soluble NH4-N (371 mg kg(-1)), C/Nsolid (18.3), C/Naquoeus (5.8) and seed germination index (GI) (66.5%) indicated that pile A achieved maturity after 49 days of composting. After 63 days of composting, pile B contained 5352 and 912 mg kg(-1) of DOC and soluble NH4-N content, respectively, which was much higher than the criterion of 5% and 400 mg kg(-1), indicating its immature nature. Pile B showed a relatively low GI value of 46%, which may be due to its high indigenous EC value as a result of larger amount of pig manure. Therefore, co-composting of pig manure with sawdust at a low initial C/N would require a composting longer than 63 days, and, the high salinity due to the large amount of pig manure would pose a potential inhibition on plant growth.

Thanks everyone