In several rooms I have a wall switch near the door that switches an outlet, and a light/fan fixture operated by a pull chain. I would like to be able to switch the light fixture from the door switch, and have the outlet be always on. What I want to do is the following: Wire from the fixture to the existing switch (external to the wall, under a wire tunnel to minimize the work). At the switch, directly connect the wires from the socket to each other so the circuit is always closed. Is this safe? Is there any reason not to do it this way?
For the socket, yes, simply take the 2 wires off the switch and splice them to each other. This will "free up" the switch for other uses.
Now if it's a split receptacle (one outlet constant-on, one outlet switched) the simple matter is to do as the above. However, inside the outlet, will be 3 or 4 wires feeding the outlet. Depending on how it's wired, it may be possible to "tidy up" by bypassing the wire-run between the outlet and the switch. How exactly to do that will depend on how it's wired, diagram it out (or snap photos) and ask again here.
To switch the fan, if your plan is surface mount conduit, that is easy. Code requires that you use proper, rated actual conduit, not the cheap wire-mold meant to guide USB cables. Run individual wires (THHN) in the conduit - and you get to choose your colors, e.g. black for always-hot, blue for switched-hot to the fan, yellow for switched-hot to the light. As implied, you can have a separate switch for each, provided your fixture can accommodate that (most can).
You will also need to wire a ground which must be green. White and gray are reserved for neutrals, and code technically requires you run one for potential future smart-switch use, but I argue, you're in conduit, it's easy to add.
First you will need to find out if the switched outlet circuit is powered at the switch or at the outlet. Then you will know whether to run your surface conduit from the light to the switch or from the light to the outlet.
You should also find where the light power cable originates. In an older home the light may be powered from the same place as the switched outlet. In that case you may not need the surface conduit.
Be aware that most electricians nowadays think it is a good idea to put the outlets in a house on 20A branches and the lights on 15A branches. If your house is wired this way, moving so many lights onto the outlet branches will obliterate the original electrician's load calculations.