How does the Supreme Court go about deciding how to rule on cases?  More specifically, how is it determined upon which Constitutional article or amendment will be at issue in such cases?

Is it merely decided upon by how the plaintiff or defendent wishes to bring up the challenge or defense, or does the Court also consider other issues that may be related.  For example, if the issue is with firearms, it could be affected obviously by the second amendment, but it is also regulated by the Constitution's commerce clause.

Quoted: How does the Supreme Court go about deciding how to rule on cases? A minimum of 4 justices decide (vote?) to have the case reviewed. More specifically, how is it determined upon which Constitutional article or amendment will be at issue in such cases? This would be the job of the petitioner, IIRC. Is it merely decided upon by how the plaintiff or defendent wishes to bring up the challenge or defense, or does the Court also consider other issues that may be related.   Yes to both, IMO. Many times the USSC has issued rulings that affected more than one amendment/law with a single case.

Much of the time the Supreme Court seeks to resolve differences in opinions given by lower courts.

To add a complication, most of your questions have answers that vary based upon what era you're discussing.

Most of the last decade was a profoundly limited (don't want to mis-use conservative) era, with the court cutting very narrow decisions tailored to the specific cases at hand.  Not Brown v. Board of Ed.

How the court decides.......There are two answers I could almost legitimately give.

1:  As the highest court of the land, they take into account precedent, lower court rulings, the applicable laws, statutes and findings, and their own political views, and find an appropriate resolution.

2:  They make it up.

Previous courts went with option #2 frequently.  This one avoids it.

Depends on how much land developers are willing to pay.

They decide how they're going to rule in a case in advance, and then find a way to rationalize their decisions through creative wordsmithing.