Congress can ordain and establish inferior courts, and can regulate the appellate jurisdiction of the Supreme Court. However, it cannot regulate much of the judiciary outside that. For one, it cannot expand the Supreme Court's original jurisdiction. Marbury. Nor can it dictate how the court applies the law. Klein. Congress's power over the federal judiciary is constrained and there is no basis in the text of the Constitution or in any case law to support your position.
However, I think a variation of the idea would be constitutional: within the Supreme Court, Congress creates two divisions for purposes of appellate jurisdiction cases (they would stay unified for original jurisdiction cases to avoid a constitutional problem). Justices would be assigned to only one or the other, no overlap. One division handles constitutional appeals, one division handles other cases.
Congress could also remove the appellate jurisdiction of the court to hear claims that laws are unconstitutional, leaving the only appellate jurisdiction as claims that officials are acting illegally or unconstitutionally or disputes of lower-level decisions as incorrect.
I'm not saying this is necessary good policy, but it would be constitutional.
Yes, it does. Jurisdiction literally means the ability to say what the law is. There is no plausible definition of appellate jurisdiction in the Anglo-American legal tradition that would not involve the power and duty to reconcile incompatible sources of law.
> Congress could also remove the appellate jurisdiction of the court to hear claims that laws are unconstitutional
No, it cannot. First, the power to make "exceptions" does not imply the ability to completely remove power. Second, whatever power Congress had under Article III was limited by later amendments, e.g., it cannot remove jurisdiction in a way that results in an impingement on the right to due process under the Fifth Amendment. There is nothing in the text of the Constitution nor in any case law that would support your implicit contention that Article III is somehow immune to being subject to the amendments process.
Third, although Klein has been a bit disfavored lately, it was reaffirmed in Patchak only a couple years ago. Some commentators focus on the plurality opinion while ignoring the fact that they're only a plurality and not the dissent because of Ginsburg's and Sotomayor's concurrence. That opinion interpreted Congress's intent generously and reasoned that Congress merely amended its waiver of sovereign immunity rather than dictated a result in a particular case. Of note, the concurring opinion explicitly agreed in principle with the "dissent," and they still ruled substantively on the merits notwithstanding Congress's attempt to strip jurisdiction.
If you'd like to continue this, please read Chemerinsky's excellent hornbook on federal jurisdiction first. And, you know, maybe go to law school.
I'm not disagreeing with the other examples you gave, including about original jurisdiction - clearly separation of powers and the constitutional structure are both real and binding. But equally, those concerns don't prevent the kind of personnel regulation we're discussing.
All money spent by the federal government in all three branches, including on judicial clerk compensation salaries, is determined by Congressional authorization and congressional statutes regarding employment and compensation. Congressional statute law heavily regulates hiring throughout the executive branch, despite that being a separate branch of government from Congress. There's no reason the judiciary is any different.
What's more, while the procedural rules of the federal courts are mostly set by the federal courts themselves, this is done under the authority of an act of Congress, which has been amended several times both to increase and reduce the courts' self-regulatory authority. As for the Federal Rules of Evidence, Congress refused to allow the version of those which the Supreme Court originally approved to go into affect, and then they later passed a substantially altered version as an Act of Congress.
The Constitution itself makes explicit Congress's ability to regulate the Supreme Court's appellate jurisdiction (which applies to almost all of its docket) and even to reduce the scope of that jurisdiction. And it has pretty plenary power to add, remove, and restructure the inferior federal courts, aside from whatever tenure protections apply to individual judges who have not been impeached by the House and removed from office by the Senate.
And, as others have said, the Senate even individually approves federal judges as per its constitutional mandate under the appointments clause. This is certainly not a separation of powers violation.