There are a number of different mechanics at work here....
This is used to represent basic background knowledge a character may have because of where they were brought up, what they do for a living etc. This is not a skill, and is checked with a simple Smarts roll. Depending on the exact character background, a modifier can be added to this roll if they are particularly likely/unlikely to know the information in question.
This is covered by individual Knowledge skills such as Knowledge(Chemistry), and represents a deeper specialization in a particular area of expertise. You need to be careful when adding these to your game to ensure that they will have significant impact and get used regularly in whatever setting you are playing in. One guideline is that if a specific skill isn't important enough to be rolled at least twice per session, it should be handled by Common Knowledge instead.
These are all the other skills that represent actually doing stuff.
So taking your particular examples...
A doctor with Healing who is using a scalpel as part of trying to heal someone just makes a Healing roll. If they are doing something more trivial under little/no pressure and without any other special circumstances, I would be usually assume they just succeed. This is particularly the case where there is no interesting consequences for failing, but then that is a good general rule for deciding whether to roll at all. A professional doctor would also probably have Knowledge(Medicine), representing their theoretical knowledge of the subject.
Establishing whether a doctor PC knows a particular doctor NPC can be done in a number of ways depending on what you as GM want to achieve. If its interesting/important to the plot, then just say they do, no roll required. If you think a roll is appropriate, I would use Common Knowledge in this instance, as this would represent their background knowledge from their job. If they had Knowledge(Medicine) as well, I might allow a roll on that with a +2 bonus for being more specialised. To increase player buy-in to this whole process, once it's been established that they do know them, you can ask the player how this is. Getting players to answer questions like this encourages them to fill out their character's back story, and is good at ensuring the PC's are strongly linked into the fictional world.
My answer for the sailor character would be the same as for the previous example.