You have asked two quite different questions there. I’ll address the second part first (where Rob argues for the decision in Bob’s favour to be overturned in his response to Bob's appeal), because it’s easier. For both parties to partially succeed at trial, and both parties to contend on appeal that they should have been wholly successful, is not that unusual. It is usually called a cross-appeal in Commonwealth jurisdictions, or in the Supreme Court of the United States, a cross-petition.
An example from the United Kingdom: in R (on the application of Carmichael and Rourke) v Secretary of State for Work and Pensions [2016] UKSC 58, the court dealt with four appeals arising out of challenges to the so-called bedroom tax. In one case, the claimant succeeded on one of two grounds, the government appealed, and the claimant cross-appealed on her failed ground:
The Rutherfords succeeded in their claim on the ground of disability discrimination. [Ms A’s] claim succeeded on the grounds of sex discrimination. Both A’ and the Rutherfords’ Equality Act claims were rejected. The Secretary of State appeals the Court of Appeal’s decision to allow the Rutherfords’ and A’s discrimination claims. A cross-appeals the rejection of her Equality Act claim … The Secretary of State’s appeal in respect of A is allowed, and A’s cross appeal is dismissed, by a majority of 5 to 2.
The first part of your question is more difficult, which is why you have received two conflicting answers. Usually appellate courts have the formal power to go beyond the issues raised by the parties, as pointed out by @Dale M. In any event, if an apex court exceeds its powers, there is no further court of appeal to say so. However, as @ohwilleke points out, to decide the case on a ground not argued violates due process. When an appellate court does this, it is said to act “on its own motion” or sua sponte.
As @ohwilleke says, in the United States, appellate courts may be able to affirm the trial court’s decision for any reason supported by the record, even if not raised by any party. This is the “right for any reason” or tipsy coachman doctrine. However, your question asks whether a court can overturn part of decision not appealed against. That is, by appealing his partial win, does Bob run the risk of the court ruling sua sponte that he ought to have suffered a complete loss?
This is discussed in Miller, ‘Sua Sponte Appellate Rulings: When Courts Deprive Litigants of an Opportunity to Be Heard,’ 39 San Diego Law Review 1253 (2002). In the introduction, Miller observes:
In fact, some of the Supreme Court's most famous opinions decided issues not presented by the briefs or addressed below. In Erie Railroad v. Tompkins, the Court overturned sua sponte an ancient precedent on applying the common law in diversity cases. Mapp v. Ohio overrules a prior case and applied the Fourth Amendment exclusionary rule to the states, without briefing or argument on the issue. In Washington v. Davis, the Court decided that Title VII standards did not apply to constitutional discrimination, even though the parties had agreed that they did. Younger v. Harris prohibits injunctions against pending state court criminal cases, even though the issue was not argued on appeal. Indeed, in Stanley v. Illinois, the Court held that due process requires hearings and an opportunity to make submissions before a state can terminate the parental rights of unwed fathers. But the Court decided this without briefing or argument – without a hearing on the issue or an opportunity for the parties to make submissions.
Most of those cases involve a petition for certiorari in which the petitioner got more than they asked for. However, Younger v. Harris is closer to the Bob/Rob scenario. Younger, the District Attorney of Los Angeles County, prosecuted Harris under the California Criminal Syndicalism Act. Before the criminal trial, Harris sued Younger in the Federal District Court, which held the Act void for vagueness and overbreadth, and enjoined the prosecution. Younger appealed to the Supreme Court, but rather than determining whether the Act was constitutional, the Supreme Court reversed the district court’s decision on the ground that federal courts should only enjoin state criminal prosecutions in extraordinary circumstances. So, Younger “won,” but Harris’s constitutional argument remained open and supported by the district court’s opinion, which Younger could not challenge again without first securing a conviction at trial.
To summarise, appellate courts sometimes do overturn decisions based on arguments not raised by the parties. However, this is exceptional and is generally considered procedurally unfair. If an appellate court thinks the parties have missed an important legal argument, it is more common and appropriate for the court to raise it at the hearing, and invite the parties to consider and make submissions about it.
