Air carrier First Officer reported glide slope anomalies and autopilot failure to capture the localizer during approach resulted in steep descent and an aircraft terrain warning. The flight crew disengaged the autopilot; conducted a go around and completed the approach.

Narrative

We were on the ZZZZZ1 arrival into ZZZ and set up for the ILS Z XXR. After passing ZZZZZ2; we were told to fly present heading and descend and maintain 3000 ft. MSL. As we approached the river; I began to slow to 190 KIAS and asked for Flaps 8 because I felt we were being vectored in close to ZZZZZ3. We asked approach if we could get lower because we were being vectored in between ZZZZZ4 and ZZZZZ3 at 3000 ft. Approach told us to turn to 020; descend and maintain 1800 ft. MSL; cleared for the VISUAL XXR approach. I set 1800 ft. in the altitude select and began my descent while I turned to 020. Because we were cleared for the approach; I activated Approach Mode. The LOC and glideslope (GS) captured immediately on my side. I thought this was strange because we were still intercepting the LOC; and we were above the GS. The aircraft turned to approximately a 050 but did not appear to capture the LOC even though the Flight Mode Annunciator (FMA) showed captured. The aircraft also began to pitch nose down and began a steep descent. The Captain (CA) announced I needed to get under the GS for it to capture because his side still showed LOC and GS armed; not captured. As I was trying to diagnose the problem; the airspeed began to rise. The CA announced it and deployed the Flight Spoilers to arrest the descent. I tried to use Vertical Speed mode to shallow the descent; but it did not work. At approximately 235 KIAS; the flap over speed clacker was heard. Shortly after; the 'Glideslope' alert was heard then the 'Terrain; Terrain; Pull Up' was heard as the autopilot was disengaged and a go around was initiated. The CA contacted approach; and they told us to turn to 360 and climb to 2000 ft. We were vectored around and tried the approach again; entering from the north. Approach told us to turn to 130 and intercept the XXR LOC. I engaged NAV mode; and the autopilot did not turn in time so we began to fly through the LOC. I engaged Heading Mode and corrected the course. Once on-course; I reengaged NAV mode and the LOC captured successfully. We were cleared for the Visual XXR a second time; and I engaged Approach mode; and it captured the GS successfully. Once on the ground; we contacted maintenance about the flap over speed; and the Autopilot anomalies. Maintenance decided to defer the Autopilot; and we remained on the aircraft for 2 additional legs (ZZZ-ZZZ1 & ZZZ1-ZZZ). On the next leg; we flew the RNAV XY into ZZZ1; but experienced similar LOC and GS issues coming back into ZZZ on XXR. The CA flew ZZZ1-ZZZ and was on the ZZZZZ1 and ILS Z XXR. As we approached the XXR LOC from the south; we noticed the GS indicator was fluctuating almost full scale up and down. He was given a heading to intercept the XXR LOC in between ZZZZZ4 and ZZZZZ3; and with just the Flight Director; the aircraft started to fly through the LOC again. He caught it and flew raw data until on course. Once cleared for the approach; he engaged approach mode and was able to capture the LOC and GS. At first; we thought the gusty winds made it hard for the autopilot to capture the LOC and GS. The winds were 040/14G23 initially and 050/20G27 when we landed. However; when the CA flew his leg back into ZZZ; the winds weren't as bad; and the autopilot wasn't used. There were still large fluctuations in the GS but never any red flags or missing data. The Flight Director was displaying inaccurate data which caused the autopilot to make the wrong inputs. After landing back in ZZZ the second time; the CA called maintenance and decided to upgrade the MEL due to safety of flight. If this same situation had happened in IMC; it would have been much harder to identify as opposed to under VMC. I think the LOC and GS capture was a false capture. Because I devoted too much time trying to diagnose why it was a false capture; I didn't observe the rapid descent. Once I noticed something was wrong; I should have disengaged the Autopilot and hand flown. This would have been safer thanrelying on automation.