Absence of cytoplasmic Intermediate Filaments in Arthropods
It is stated that out of all eumetazoans in Kingdom Animalia, only arthropods do not contain such structures.

I found this paper(1), eliciting an exception (within an exception)- Isotomurus maculatus:

Here, we report the first evidence for the expression of a cytoplasmic IF protein in an arthropod - the basal hexapod Isotomurus maculatus. This new protein, we named it isomin, is a component of the intestinal terminal web and shares with IFs typical biochemical properties, molecular features and reassembly capability.

I would suggest you to read the paper, for more information.

How is this possible? Are there any replacements to the conventional intermediate filaments present in cells?

Both electron microscopy and molecular cloning studies have been unable to detect any cytoplasmic IF protein in these organisms, although they do express a nuclear IF system made of authentic lamins. Other cytoskeletal components have been proposed to have assumed, in arthropods, the mechanical functions that are usually played by cytoplasmic IFs in other organisms. For example, wing epithelial cells are stabilized in insects by a cytoskeletal array consisting of parallel bundles of 15-protofilament microtubules and actin filaments.

Do arthropods have special cellular behaviors or signaling networks to correspond with this absence?

I am not sure if this is what you might be looking for, but the Delta/Notch signalling, which may be involved in posterior elongation in arthropods, does not require IF's.(2)

The Notch cascade consists of Notch and Notch ligands, as well as
intracellular proteins transmitting the notch signal to the cell's
nucleus. The Notch/Lin-12/Glp-1 receptor family was found to be
involved in the specification of cell fates during development in
Drosophila and C. elegans.

1- https://bmcbiol.biomedcentral.com/articles/10.1186/1741-7007-9-17
2-https://en.wikipedia.org/wiki/Notch_signaling_pathway#Mechanism_of_action