Captain America is given the same flight, durability, strength and speed as Superman, however he will lack X-Ray Vision, Freeze Breath and the other exotic powers Superman only-sometimes uses. But he keeps his shield. How will he do if he fights Superman?
Captain America in both scenarios (obviously assuming he knows how to use the powers otherwise he will be smashing through mountains like DCEU Superman at the beginning of Man Of Steel).
Steve wins both for the same reason that Superman defeated Wraith (a more powerful alien being with powers similar to his, just at a greater level, in Superman Unleashed ...a good read). In that story, Superman defeated Wraith at one point hand to hand, stating that he (Superman) often had to fight more powerful foes with no backup support, while Wraith always relied on just his powers and anyways had the full might of the US military behind him (Wraith was the secret weapon for the US military, and was allegedly what caused either the Hiroshima or Nagasaki blast in WW2).
Steve would defeat Superman for ironically the same reasons. Steve is used to fighting more powerful opponents, and has adapted his skills and fighting abilities towards that. Giving him the strength, speed, invulnerability etc of Superman would basically result in ...a Black Adam type character with the fighting abilities of Batman.
If Cap is competent in the use of his powers (and he should be, otherwise the thread premise is basically a character incompetent in the use of unfamiliar powers vs one of the most powerful characters in comics, and i doubt you'd do that), then i believe he wins, KMCer.
If Supes can sundip to become stronger and Steve can't, it wouldn't be fair. Also, I doubt that HV and FB would be a decider in the fight.
This all comes down to what the OP really want the battle rules to be. Can Superman sundip to become stronger? And can Steve also sundip to increase his strength to become stronger? Too vague fight description imo. Should have been more precise about that.
Do you forget that in this fight, Steve have the same physical abilities as Superman. I would assume with speed he means that he can do anything Superman can with his speed. So their perception of things should be equalized.
FB wouldn't really matter for Steve since you know, he can and will avoid it when Superman uses it. And HV would also be useless against Steve because he can block it with his shield. This will come down to a h2h fight (and if Superman isn't allowed to sundip), Steve would beat Clark.
The OP really should not allow Sundip, and if he does, Steve should also get it. Because otherwise it's just an unfair fight. And it would all be a pointless thread.
If Steve have learned to utilize his powers to the fullest, and Superman doesn't sundip to increase his speed and strength and durability, Steve would win simply because he is a better and smarter fighter than Superman. If Superman is allowed to sundip,, and Steve have not learned to control the powers he have, he would most certainly win. HV would do jack shit since Cap have his shield. As for FB, wouldn't really matter as Steve can avoid it due to his increased speed.