Cat5e cable is capable of running a gigabit ethernet home network. For the telephone jacks you've posted in the photos only wire continuity is important; telephone is on a different level than ethernet, as far as wire consistency and specifications go. You'll have to make sure as you re-terminate each cable that proper pairs are used and are untwisted only as much as necessary.
Each cable should be a "home run" to a central point. Your punchdown panel in photo 2 looks like it may already be a suitable location, but you will likely have issues with cable length there unless there is some slack in the walls. A lot of that unjacketed cable should be cut off. If you have room, it would be good to put a small patch panel in that location - if not you can use modular ethernet jacks. You can probably fit 6 keystone ethernet jacks into a wall plate there.
On each other end at the individual jacks like shown in photo 1, a modular ethernet jack can be used with a wall plate (keystone again would allow you to keep the coaxial in the same plate as shown) and it looks like you have a little slack to work with there.
Once the jacks are in place you can connect a router or switch to each jack in the central "hub" location (photo 2) using patch cables (cat5e would be fine, as another commenter said your network connection will only be as good as your worst cable so cat7 is overkill here) and another patch cable at the wall plate ends to connect the devices.
The Cat3 you have outside the house doesn't matter, ignore that. You can remove it or leave it (but if you remove it, be sure to seal up the holes in the exterior siding).