We explore deep rest-frame UV to FIR data in the COSMOS field to measure the individual spectral energy distributions (SED) of the ~4000 SC4K (Sobral et al. 2018) Lyman-alpha (Lya) emitters (LAEs) at z~2-6. We find typical stellar masses of 10$^{9.3pm0.6}$ M$_{odot}$ and star formation rates (SFR) of SFR$_{SED}=4.4^{+10.5}_{-2.4}$ M$_{odot}$/yr and SFR$_{Lya}=5.9^{+6.3}_{-2.6}$ M$_{odot}$/yr, combined with very blue UV slopes of beta=-2.1$^{+0.5}_{-0.4}$, but with significant variations within the population. M$_{UV}$ and beta are correlated in a similar way to UV-selected sources, but LAEs are consistently bluer. This suggests that LAEs are the youngest and/or most dust-poor subset of the UV-selected population. We also study the Lya rest-frame equivalent width (EW$_0$) and find 45 extreme LAEs with EW$_0>240$ A (3 $sigma$), implying a low number density of $(7pm1)times10^{-7}$ Mpc$^{-3}$. Overall, we measure little to no evolution of the Lya EW$_0$ and scale length parameter ($w_0$) which are consistently high (EW$_0=140^{+280}_{-70}$ A, $w_0=129^{+11}_{-11}$ A) from z~6 to z~2 and below. However, $w_0$ is anti-correlated with M$_{UV}$ and stellar mass. Our results imply that sources selected as LAEs have a high Lya escape fraction (f$_{esc, Lya}$) irrespective of cosmic time, but f$_{esc, Lya}$ is still higher for UV-fainter and lower mass LAEs. The least massive LAEs ($<10^{9.5}$ M$_{odot}$) are typically located above the star formation Main Sequence (MS), but the offset from the MS decreases towards z~6 and towards $10^{10}$ M$_{odot}$. Our results imply a lack of evolution in the properties of LAEs across time and reveals the increasing overlap in properties of LAEs and UV-continuum selected galaxies as typical star-forming galaxies at high redshift effectively become LAEs.
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