You need the converted style (has to be in old format (ie: v_xxx and g_xxx); CheapoCon hasn't been updated to support the new format yet) in the same folder as CheapoCon. You don't need the style files when extracting the levels via Essman's converter. (The reason for this is that Essman's tool just needs to know "Object A, of type B, exists at position X, Y", whereas in some cases CheapoCon needs the graphic set to look up exactly what *type* of object it is too.)
And yes, it's completely normal for the objects to be missing from the large PNG file. The large PNG file is the level's terrain layout (this is how Cheapo works - the level layout just becomes a single large image, rather than being made up of smaller pieces as eg a Lemmix level would be), while the small one is the preview screen image (hence why it does include the objects).
Conversion to SuperLemmini is possible but a bit tricky at the moment - I'd hold off on that until I release an updated (or replacement) tool, but if you do really want to do it, convert it to a NeoLemmix level the usual way (but with zero passes on the terrain detector), create a SuperLemmini version of the graphic set, and use NeoLemmix Editor to convert the NeoLemmix level. For creating the special graphic PNG file, use the normal-size one from the extractor's output; you'll need to resize it (losslessly; use "Nearest Neighbour" if your graphics editor offers you a choice of resize filters) to double the size, then replace all the black background with transparent. You might also need to slightly adjust the positions of any objects that are horizontally flipped - the old NeoLemmix graphic set format required their widths to be padded to multiples of 8, and CheapoCon did account for this when converting the levels. This will mean that horizontally flipped objects are slightly off-position when using a SuperLemmini (or even new-format NeoLemmix) graphic set, where the widths can be used as-is.
I'll probably look into doing an updated tool over the next few days; quite possibly including removing the intermediate step of having to use Essman's tools.