Is paying extra for "Controlled Impedance" boards necessary?
If you specify controlled impedance/controlled dielectric, they will test your board to ensure that the traces are at the specified impedance.  In your fabrication notes on your PCB printout, specify the nets and their targeted impedance (with tolerance, e.g., 50 ohms +/- 2 ohms).

They will either test a small test strip that is manufactured on the same panel as your boards; or they will test all nets as needed.  This will help catch boards that do not meet spec, before they end up being stuffed with components.

BTW, the "weave" of the board may affect the actual impedance of any particular trace, even when the traces are built to spec (see PCB Dielectric Material Selection and Fiber Weave Effect on High-Speed Channel Routing - Altera Application Note).

Your board fabrication note should specify what the target impedance is, and on which traces those impedance values apply to.   (Example: Trace width of 8 mils shall be at 50 ohms +/- 10%.)  The fabricator may adjust your trace width slightly to meet the target impedance.