In a sequence diagram, when having a multi-instance object, would it be correct to use a loop frame around the sequence related to accessing that object?
And, is it regular usage to use condition and alternate condition frames to signify error messages and alternate flows?
On
Yes, using loop is correct (see e.g. Agile Modeling: UML 2 Sequence Diagrams: An Agile Introduction, Figure 6). You may also find useful lifeline with selector expression
As for the errors and exceptions, it is not where UML sequence diagrams are very strong. On the other hand UML activity diagrams are equipped with interruptible region and interrupting edge especially for this purpose.
uml-diagrams.org: Sequence Diagrams: Questions & Answers:
How to model exception handling in sequence diagram?
UML provides neither notation to model exception handling in sequence diagrams nor any reasoning why it is absent. Some clumsy approaches to model try-catch blocks are by utilizing combined fragments - alt (alternatives) and breaks, while adding stereotypes for reply messages representing thrown exceptions. There are several proposed notations for exception handling
Elements "frame" in your description are colled Combined Fragments in UML. For more details of how to use combined fragments read UML Superstructure document, Interaction section. It is free document. Download it from UML web
Combined Fragment are used to define conditions, loops, alternatives, concurrency etc.