"With an (instrument)" and "by (instrument)"

"We forced the door with a jimmy" isn't really idiomatic because jimmy is a word that means:

a short crowbar used by a burglar to force open a window or door

Nothing else about the sentence is being used abnormally or in a manner that couldn't be solved by a dictionary.

Furthermore, my ear doesn't consider "we forced the door by jimmy" normal usage and it sounds funny. I have never heard it before and jimmy isn't being used in a typical sense. The more common way to say this is:

We jimmied the door

As for your list, these are the sentences I find unusual or awkward:

• [1-b] Lines were drawn by straight edge.
• [3-b] The beans were eaten by chopsticks.
• [4-b] The room was cleaned by broom and mop.
• [6-b] The length of the rope was measured by ruler.

These are all fine:

• [1-a] We drew lines with a straight edge.
• [2-a] We held the unknown object with the tongs we found in the lab.
• [3-a] We ate the beans with chopsticks.
• [4-a] We cleaned the room with a broom and a mop.
• [5-a] We cut the paper with such a peculiar instrument.
• [6-a] We measured the length of the rope with a ruler.
• [2-b] The unknown object was held by the tongs we found in the lab.
• [5-b] The paper was cut by such a peculiar instrument.