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How are bishops chosen now?

The procedure, as I understand it, in the Western church (IOW, the Latin Rite, not the Byzantine Rite churches) is supposed to be:

Bishops are encouraged to collect information on priests they think would make good bishops
They then forward those names, CVs, etc., to the Papal Nuncio in Washington
When a vacancy occurs, the Papal Nuncio prepares a list of three candidates, called a terna, and submits it to the Congregation of Bishops in Rome.
The Congregation does their own investigation (allegedly this includes soliciting -- under strict secrecy -- the opinions of laypeople who know the man being considered), and then selects a candidate from the list and submits his name to the Pope.
The Pope can approve the nomination, or request another name, causing the process to continue, etc.

Apparently PF has seen fit to circumvent this process in some cases.

Whats the difference between a bishop and an archbishop?

An archbishop is the bishop of a "metropolitan see," which has lesser dependent sees called "suffragans". A metropolitan see generally belongs to a major city where a lot of Catholics live ... St. Louis, Philadelphia, etc.

And how do some bishops or archbishops get designated as cardinals in the church?

The Pope appoints them. Some sees generally are expected to have their bishops raised to the rank of Cardinal, e.g., it was considered odd and irregular when PF did not name Chaput of Philly a cardinal but did name Bad Tobin of Newark one. Newark has not traditionally been a cardinalatial see, while Philly has been.

Is there a systematic problem with how people are recruited and promoted in the church?

Before we go there, I think we need to address the issue that PF is apparently circumventing the system put in place and developed over centuries, in order to put his allies in positions of power, especially in the College of Cardinals.