  Narrow-line Seyfert 1 (NLS1) galaxies are believed to be active galactic
nuclei (AGN) in the early stages of their evolution. Some dozens of them have
been found to host relativistic jets, whilst the majority has not even been
detected in radio, emphasising the heterogeneity of the class in this band. In
this paper, our aim is to determine the predominant source of radio emission in
a sample of 44 NLS1s, selected based on their extended kpc-scale radio
morphologies at 5.2 GHz. We accomplish this by analysing their spatially
resolved radio spectral index maps, centred at 5.2 GHz. In addition, we utilise
several diagnostics based on mid-infrared emission to estimate the star
formation activity of their host galaxies. These data are complemented by
archival data to draw a more complete picture of each source. We find an
extraordinary diversity among our sample. Approximately equal fractions of our
sources can be identified as AGN-dominated, composite, and host-dominated.
Among the AGN-dominated sources are a few NLS1s with very extended jets,
reaching distances of tens of kpc from the nucleus. One of these, J0814+5609,
hosts the most extended jets found in an NLS1 so far. We also identify five
NLS1s that could be classified as compact steep-spectrum sources. We further
conclude that due to the variety seen in NLS1s simple proxies, such as the star
formation diagnostics also employed in this paper, and the radio loudness
parameter, are not ideal tools for characterising NLS1s. We emphasise the
necessity of examining NLS1s as individuals, instead of making assumptions
based on their classification. When these issues are properly taken into
account, NLS1s offer an exceptional environment to study the interplay of the
host galaxy and several AGN-related phenomena, such as jets and outflows.
[Abstract abridged.]
