You can buy odourless turpentine if you want 🙂 It costs more of course, but there's no smell. Regarding underpainting, normally you would thin the oil paint down with turps or white spirit so you can apply it in washes... and how you use it might depend on what final look you want from your painting. Traditionally the majority of Renaissance to Baroque underpaintings were made 'en grisaille' - just painting in shades of grey, from white to mid grey. Once the underpainting was dry, coloured oil paint mixed with linseed oil would be applied on top in transparent glazes, so that the shading from the underpainting tinted the colours above to form the final result - if you look at the Flemish masters (Jan Van Eyck and onwards etc) that's how those paintings were made. Some artists, especially after the Renaissance used an underpainting in shades of brown or used sepia ink. This tends to give a 'warmer' colouration to the overpainting, but can cause issues if you have areas of blue and green in the final artwork which tend to come out somewhat muddy looking. The old advice (which is true) if you want your oil painting to last / not crack was to work 'fat over lean' which meant the underpainting oil paint was mixed with turpentine to thin it down (so it has less oil, or 'fat' in it) and then when you applied the coloured layers on top you would add more oil (or 'fat') to the paint to make it transparent. That way the oily layers are on top of a stable base that doesn't shrink - the oil layers tend to shrink as they dry (OK oil paint never really ever dries, but you know what I mean), so they need to have a stable base beneath them that doesn't shrink, and then are given enough time to feel touch dry before the next layer is applied.
This is of course a slow way of working; I used to keep several paintings on the go at the same time so that I would add layers to each before waiting for each layer to dry before adding the next layer... and it all went out the window with the Impressionists! Alla Prima is a much faster way of working.