Even within states, the court procedure and certain court rules can be substantially different between counties/jurisdictions. But from state-to-state, there is a huge variance.
In Pima (Tucson and surrounding area) and Maricopa (Phoenix and surrounding cities) County, AZ, for felony charges, a plea offer extended by the prosecutor’s office is only available at a certain, somewhat later stage of the proceedings, but still well before trial. That plea offer usually, but not always, guarantees either a lower level felony than the defendant is charged and/or a fixed sentence (probation, probation with a certain amount of jail time, a certain prison range with the range depending on the class of felony, or a fixed prison term). In these jurisdictions, the plea offers often times also come with an expiration date, usually at least a month or so before any trial or final hearing date prior to trial. Someone could try to plead guilty straight-up to their charges at an earlier stage in the proceeding, but the judges wouldn’t be very prepared to handle that and pleading guilty straight-up would essentially be the same as going to trial and losing, only without any possible guarantees or punishment ceilings a plea offer might have.
The Sixth Amendment has been interpreted to give defendants a right to a trial by jury for all offenses where they may be incarcerated in jail/prison for more than six months. Despite the clear “in all criminal prosecutions” preceding language in the Amendment. Some states go further and provide a right to a trial by jury for either whole classes of misdemeanors or for certain misdemeanors where the maximum sentence of incarceration someone could receive would be six months or less. For example, in Arizona, a DUI (driving under the influence) is a class 1 misdemeanor (the highest of the three classes of misdemeanor) and even a first time offender convicted of a DUI must, by law, spend 24 hours in jail with another 9 days that can be suspended if the defendant successfully completes an alcohol abuse education class. The maximum sentence a defendant can face for a class 1 misdemeanor in AZ is exactly 6 months in jail, but because of the mandatory jail time imposed by law upon conviction, the AZ legislature has made DUIs a specific offense where a defendant has a right to a trial by jury.
The factors that go into a defendant and their attorney deciding on whether they want to go with a trial by jury or trial by judge alone are complex, but the default setting (where there is a right to a trial by jury) is generally a jury trial. Judge-alone trials usually might be done if there are complex legal issues or arguments that might “turn off” or strike a jury as tricky or disingenuous or there are particularly bad facts/photos/videos/etc. that might unduly sway a jury but have less of an impact on a judge and there is still a good factual or legal argument that can be made despite the bad or damaging materials.
(In the military justice system, this issue is even more complicated since a military panel, which is roughly kinda/sorta comparable to a civilian jury, does not have to vote unanimously on guilt and prior to 2017, only had to vote 2/3 in favor of guilt for a conviction to occur. Also, again prior to 2017, if an accused service member picked a trial by military panel, the military panel would get to decide the sentence upon a conviction. In the civilian world, outside of death penalty cases where a jury has to unanimously recommend a death sentence following a conviction, civilian juries generally do not have a role in sentencing.)
As for the last question, generally, if charged by the same sovereign, someone cannot be convicted in two separate cases for the same criminal conduct. That would generally be double jeopardy and subject someone to double punishment. For example, someone could not be charged with misdemeanor assault (which in Arizona is a combo of assault and battery) for punching someone in the head, the person later dies of that single punch a day later, and then charged in a different court with felony manslaughter by punching someone in the head, and have both cases proceed to trial. The misdemeanor charge would be dismissed so that the felony charge could proceed without complication.