Air carrier Captain reported severe weather conditions; problems with airspeed / altitude control and mechanical problems. These issues led to a loss of control event; FAR speed deviation; and FAR altitude deviation.

Narrative

Too many threats to count. Blizzard; LLWS; maintenance; delays; looking forward to a max duty day; proficiency; 2 jumpseaters.Weather was winter-nasty in ZZZ. Jet was a bit late arriving from ZZZ1. Weather enroute to ZZZ2 was a fast moving strong front. We were already delayed before we left hotel; but arrived as scheduled to airport. Upon arrival at jet we received ACARS EDCT mssg. Overall late from schedule by about XA:30. On preflight we discussed clutter; deicing; LLWS. Considering conditions we decided to use F5 instead of a higher flap setting. It was a Max Thr light jet and we discussed the differences between flap settings for clutter ¼' slush; and LLWS. I was concerned with ovsd the flaps on a MAX TH climb out so we went with F5. Most of slush was at far end of Runway XXC. Any delayed rotation for LLWS would put us in the slush; so that was the decision matrix. Also planned to call for FLCH at 800 ft. AGL due to a low level off of 3000 ft. MSL. Maintenance-wise; we had INOP RT IRS so FO (First Officer) had some switches to put in non-normal positions. Also; no Wi-Fi so we knew we would have to go old-school for the [storm] mitigation in ZZZ2. Otherwise; Preflight was normal. We ran all checklists and called for push. Just as 2nd engine was starting a RUDDER RATIO EICAS came up. Entered ELB entry and FO called ZZZ Maintenance then looked up malf in QRH. I referenced the MEL to see what might be deferrable. A RUDDER RATIO EICAS isn't deferrable. Either way; Maintenance came on VHF 2 and had us reset about 8 CBs. Took a while; but EICAS finally cleared. We called for further taxi to Deice pad. While deicing we talked about weather conditions; LLWS; light weight; MAX TH again. We were good to go. Finished deicing; ran all checklists again to make sure everything was setup properly. Took it slow to set the tone. Also; to give me a few extra minutes of brain time. I'm returning to the [aircraft type 2] after 18 months from [aircraft type 1] Captain on reserve. I hardly flew at all in Year. I was/am not very proficient from a flying standpoint. Here we go... Short taxi from Deice pad to Runway XXC. Only a few departures ahead of us. No PIREPS from Tower. We held short to send the FO back to check the wings. They were using Type 1 in the pad; and the HOT had already expired before they even finished spraying. At this point the Vis was about ¾ miles and very light snow; not really an issue. He came back; reported All Good; and we finished checklist; called Ready. Lined up well on centerline; cleared for Takeoff. Power up to MAX. Wind was something like 340/19G33 and I was impressed with how well I stayed on CL. Rotated on speed. About 800 ft. we started to encounter moderate thunderstorms. I called for FLCH CMS. Ride was really rough and AS was jumping around but manageable. Called for flaps as soon as I could. The FO commented ALT CAP pretty much as soon as he pressed FLCH; about 900 ft. AGL. With MAX THR we were climbing aggressively. Flaps came up; slowly. At the F1 call; about 2000 ft.; the thrust; which was back near idle; came out of my hand and went to CLB THR. AT were still engaged; but I pulled them back to keep from overspending the flaps. AS jumped from CMS 223 to bottom of redline; 250 for FL1. I held the throttles back as I was attempting to level off at 3000 ft. Flaps came up and there was no overspeed. It was extremely difficult to hold AS and ALT in the turbulence. It was probably severe. But what was way more difficult to manage were the Autothrottles. They kept wanting to go to CLB. At this point I was fighting the AS and ALT in the turbulence and the AT. My scan had reduced to a pinhole; concentrating on the very basics; so I wasn't watching FMAs. Way too dynamic here. I finally disconnected the AT; engaged the C AP; then engaged the AT. AS was probably 300 by now. Immediately the AT went to CLB. TB still probably Severe. Clicked everything off again and finally was able to recover at 4000 ft.; 1000 ft. high; and maybe 320 KIAS. Thunderstorm was moderate by now but not near as bad as before. I had FO [advised ATC] now. He stated we were having difficulty maintaining alt and AS. ATC asked if we needed assistance or if we were returning to ZZZ. At this point the jet was becoming more manageable. We then told them we were ready to continue to destination. We rescinded the [priority handling] and continued to ZZZ2. Enroute we all discussed what happened and what we saw. We came to the conclusion that it was a horizontal Windshear-type event. But none of us could figure out why the jet was calling for CLB THR. There were never any overspeed warnings or stick shakers; but FO noted he saw a FLAP LIMIT FMA. I didn't see a single FMA as I was trying just to keep the plane upright. At that point in the climb from F1 to flaps up the speed was getting most of my attention; and I kept it out of the red. FO and I are convinced the FMC was programmed properly. Upon arrival in ZZZ2 we had ZZZ2 Maintenance do an AT test and nothing came up. That solidified our wind shear call. There were no Cautions or Warnings and no Windshear callouts. Looking back; it was very startling. Happened very quickly. Max thrust climb in a light jet; blizzard; slush; LLWS; poss severe turbulence; proficiency: it all added up. During debrief we decided that this was an Upset situation. We were doing the procedures but nobody called it. We unintentionally violated SPD and ALT having gotten to about 320 KIAS and 1000 ft. above level off (4000 ft. vs 3000 ft.). It took all of my skills at that moment just to get it there. There were no EICAS; Warnings or Cautions; Stick Shakers. Only a FLAP LIMIT FMA. It was ugly; quick; and violent. I felt we discussed every possibility up to that moment. In retrospect; the only thing we could have done differently was call UPSET. Not sure it would have made a lot of difference; but the AT would have been more of a priority in the recovery. We did disconnect them early in the event; but re-engaged them too early in the recovery. They fought us until we were at 4000 ft. Should have kept them off at first decision point. My scan was minimal so I don't have comments about much; only what I was focusing on; included here. Called [authorities] and Chief Pilot after landing to debrief. Chief Pilot said they thought it went well considering the events and gave us a 'Good Job' report.