If I start writing my PHP code in editor I see proposals from picture #1 - unsorted methods contains a half-written expression (inconvenient)
If I write the same expression and press Ctrl+Space I see expected proposals from picture #2 - methods sorted by the expression beginning (convenient)
How to ensure that when entering an expression, the list of proposals corresponds to picture #2? Coding will go faster if I do not need to press the Ctrl+Space combination
Is there any Eclipse setting to fix it or is it a bug?
You haven't webkit installed on your linux. See for example this article how to do it: How to get Eclipse SWT Browser component running on Ubuntu 11.04 (Natty Narwhal) with Webkit?
Note since upcoming (Photon 4.8) release, webkit 2 will be default.