A good candidate for this type of study is factorial ANOVA (Analysis of Variance), using independent variables:
technique (4) and
The dependent variable would be the difference between each technique and the intra-arterial measurement.
Overall, the most accurate measurement would be the one with the smallest difference...
After examining the ANOVA for significant effects, you could then do some post-hoc comparisons along the lines of your research questions that you listed.
The advantage of doing ANOVA is that you can look at interactions between the factors - i.e., do doctors/nurses make more accurate measurements on certain techniques; do certain techniques work better on children than on adults, etc.