How do I run wired internet from a single router to several different rooms in an apartment?
This is my recommendation and similar to what I have in my home.

Since you have the walls open, this is the perfect time to do this.  Don't limit yourself to what you need now, think about the future.  With the push for streaming and internet connected devices, almost every room will probably have at least one internet connected device.  I recommend putting drop points anywhere you plan on having a desk, tv or phone.  Also put drop points where you think desk or tv's would be nice.

As for cable, Cat6 will do but I recommend using Cat6A (or sometimes called Cat6e).  This cable will handle 10gig+ so in the future (5 years) when 1gig is not enought you will not have to pull new cable.  Though at 10 meters, even Cat6 will handle 10gig, but my mindset is put the best in the wall since it is so hard to remove.

As for installing the cable, no sharp bends and no stretching of the cable.  Similar rules for coax cable.  Most installers now how to install that, just tell them to treat the install the same.

As for the drops.  Have the installers either put a one gang electrical box or a low voltage box where you want the jacks installed.  For the jacks, here you can save a little money and use Cat5e jacks.  If you need more bandwidth in the future, these are easy to upgrade to Cat6(a/e).  They have either plates with a jack integrated or whats called a keystone plate.  With a keystone plate, you add modules for what you want into the plate.  Modules can include telephone, coax, audio or ethernet.  Personally in my house I use the keystone plates and most rooms I combine all the low voltage requirements in one box (most being coax and ethernet).

For the other end of the run, I personally would have them go to a closet or cupboard that has power.  The router does not need to be where the cables all run to and I find this to be a cleaner solution.  In the closet, place a network switch to handle connecting all the devices.  Also the network switch does not have to be as big as all the drops in your house, only as big as the drops that are currently being used.  I'm assuming you want the router next to your TV where your coax is.  Just add a drop there and pug your router into it.  The switch in the closet will handle splitting it to the rest of your house.  And this will also give 3 or so ports for the devices around your TV.  This is how it is setup in my house and all 3 of those extra ports are used.  One for the TV, BD Player and Media Streamer.

As for the closet where the end of the run is, if you only have 6 or less drops you can use a keystone plate with ethernet modules.  If you have more than 6, I would recommend putting in a patch panel.  They come in sizes as small as 8 ports up to any size you would need for an apartment.  And with short half meter patch cables, connect the panel to the switch (which can be hung on the wall near the panel).