■ I was conducting a CFI single engine add-on with a local Designated Pilot Examiner (DPE). [The airport] was extremely busy and using Land and Hold Short (LAHSO) rules for Runways XX and XY. We were probably on the ground for…thirty minutes before we were cleared to taxi…and another thirty minutes before we were cleared to takeoff.

For the final maneuver of the check ride, the examiner asked for a power-off 180 landing to Runway XX. He was directing me throughout the entire maneuver (when to turn base, what airspeed to hold, how much bank to use, when to deploy flaps, etc.). His instruction led me to a very high final approach altitude. I stated, “We need to go around. We are too high.” I was scared we would break the hold short lines for Runway XY. He replied, “We are not going around. I don’t want to get stuck up here for another hour.”

■ We took off with 13.0 fuel on board. During climbout at approximately FL180, we received a fuel imbalance yellow [indication] on the Number 1 Fuel Gauge. According to the gauge, fuel was burning out of Number 1 Tank at a high rate. It seemed double the rate as usual. Fuel flows were the same on both engines. Fuel burned matched and were equal on both engines. Total fuel remaining equaled the planned fuel remaining. We even had a flight attendant check the wings for a possible fuel leak. We chose to run the IMBAL QRH [checklist].

With the Crossfeed [Valve] open and Number 1 Pumps off, we were able to slow the fuel burn out of the Number 1 Tank, but were not able to stop it. We called Maintenance Control over the radio and were advised to follow the QRH. At that point we had approximately 5.3 in the Number 2 Tank and 2.2 in the Number 1 Tank.

■ Flying into ZZZ, we were told there were windshear and windshear conditions reported. The Captain was the Pilot Flying (PF), and I was the Pilot Monitoring (PM). Upon being told about the conditions, I put the continuous ignition on and briefed the Captain that if we get a WINDSHEAR warning, we would go around and fly to the published missed approach point. Once cleared for the visual, the aircraft was stable at 1,000 feet, where I made the 1,000 foot stable callout. Around 500 feet the WINDSHEAR aural alarm went off with the warnings displayed on the PFD. I said, “Are you not going to go around?” the Captain said, “No, that’s a caution,” and continued…

■ Between 10 feet and 50 feet AGL during takeoff rotation, two Canadian geese flew across our nose with about 120 degrees right aspect angle. I saw the birds and attempted a very slight upward rotation to avoid them; however, impact occurred shortly thereafter into the Number 1 Engine. I elected to leave the landing gear down and the takeoff flaps set, as I did not know if the birds hit the nose or nose gear. The impact felt very hard, like it was to the lower front of the aircraft. I continued the takeoff climb to 1,000 feet AGL, and then started a right turn out and climb to 2,000 feet. At impact there was a very loud bang and instantaneous heavy vibration in the airframe.

Both the First Officer and I noticed a smell of burning bird in the cockpit. I reduced the power setting on the Number 1 Engine with no noticeable reduction in the vibration. I called for the Engine Fire or Engine Severe Damage or Separation checklist and noticed high vibration on the Number 1 Engine, approximately 3.3.

■ Trusting in his judgment as a DPE and giving in to the politics of student pilot vs. DPE, I continued to land.… As I thought, we landed very far down the runway…but stopped completely before Runway XY. Once in contact with Ground, they issued me a phone number for a possible pilot deviation. I and the DPE spoke to Tower on the phone. I was put in a very difficult situation between trusting my inner judgment or trusting the advice of a seasoned DPE, who denied my request to be conservative/safe and simply go around. This experience has taught me that a pilot can always go around, and I am truly sorry for not exercising that right.… I let the politics of “the DPE is always right” cloud my inner judgment and was scared to act against him. I will use this experience to teach the importance of a go-around to future pilots, and to always lean on the safe side of flight.

■ We decided the prudent action was to [advise ATC] and divert to [a nearby alternate]. We landed with 1.9 (number 1) and 5.1 (number 2) showing on the fuel gauges. We landed with no problems and taxied to the gate. FMC fuel burn calculations had us landing with the correct fuel at [destination]. We just could not stop the fuel burning from the Number 1 Tank.

■ [The Captain] continued to…land. Once on the ground,…I pulled the Captain aside and asked him to explain why he chose to continue to land when I thought…the indications that the plane was giving us were a warning, in which case we needed to execute a go-around procedure. He explained his reasoning…why he thought it was a caution.… I expressed why I thought it was incorrect, and the conversation ended.… I [later] researched that what I found…was correct,…and we should have gone around. I could have been more assertive to make the Captain go around.

■ I pulled the thrust down to idle on the Number 1 Engine and noticed little to no vibration thereafter. As the FO was reading the checklist, I made the decision to leave the engine running at idle and to transition to the Single Engine Landing checklist. I informed the flight attendants of the situation, and that we were returning to land. I did not brief the passengers, as we were very busy. I started the APU and set up for the ILS.

Tower was concerned about our flight path. I directed the FO to reply that we were turning right base to final and then prepared to land. On final we ran the…landing checklist and were losing sight of the runway due to heavy rain. On short final we saw the runway with good visibility to land. We landed and taxied clear, at which time I briefed the passengers on what had happened. We taxied to the gate without incident and shut down.

Things happened very, very fast and I rushed the QRH checklist. We missed one item: to select FLAP INHIBIT, which gave us a TOO LOW FLAPS warning on final approach. Due to our heavy weight and flaps 15 configuration, it was very hard to slow down. Due to the rain, I was thinking we might have to go around, but when we broke out, we were on glidepath and about Vref plus 20 knots, but slowing as we passed over the overrun. Landing thereafter was uneventful.