Gameboy Development Forum

The screen does render in a CRT way. The way it works at its deepest is using a pixel FIFO (a queue, if you prefer). The FIFO is filled every time it's needed (once every 8 pixels, also when sprites need to be taken into account, and so on).
Each scanline is rendered in the following way:

During Mode 2, OAM is searched to pick the 10 sprites that will be rendered on this scanline.
During Mode 3, the FIFO is first filled with BG tiles, then shifted according to the lower 3 bits of SCX ("fine scrolling"). Then, pixels are shifted out, and once the FIFO has only 8 pixels remaining, a tile row is fetched to fill 8 more pixels. The process repeats, and may be interrupted by the window, which more or less restarts the whole process but fetching WIN tiles instead. The palette is applied to each pixel when it's shifted out.
Mode 0 is the time left until the following Mode 2 starts - different things during Mode 3 (initial SCX shifting, sprites, and the window) can cause the FIFO to stall while something else completes, so that extra time is taken from Mode 0.

Parameters are sampled at different times and in different ways: for example, BGP (and OBP0 and OBP1) are sampled each time a pixel is pushed; SCY and LCDC are sampled each time a tile is fetched, SCX is sampled in a complex way.
You can reliably change anything between scanlines and it'll work intuitively (except maybe OAM DMA), but there are also things you can change mid-scanline to produce complex effects. I can provide further information if you're interested.