Daughter board Ethernet connector
100BASE-TX MDI will run over that connector with no problems at all in practice. It's remarkably forgiving - I've seen it run over D38999 connectors which have a much greater deviation from the 100 ohm nominal differential impedance and the reflections from the impedance mismatch only start causing functional problems when you try to make it go through more than about 5 mated pairs of those connectors before reaching another PHY (with tens of meters of intermediate cabling). I would not say the same for 1000BASE-T.

You may as well take some mitigating steps though:


Keep the distance between connector and PHY as short as practicable (a few inches is fine)
Pin assignments: Do something like GND RX- RX+ GND TX- TX+ GND to keep the characteristic impedance vaguely in the ballpark of what it should be and to reduce crosstalk. You can replace any of those GNDs with a power supply rail voltage if desired, or omit if it's at the edge of the connector.
Put the magnetics on the motherboard, avoiding hassles with circulating CM currents. But if that can't work due to space constraints, it'll almost certainly be fine anyway.
Since the connector you pictured is board-to-wire rather than board-to-board, keep the cable you use with it as short as practicable and use twisted pairs for RX and TX if it must be more than a few inches long.
Don't expect to get the full IEEE 802.3 100 meter network segment length out of it. Connect the daughterboard to a NIC or switch with a >= Cat5 patch cable no more than 10 m long.