1. The style choice is obviously up to you, but the original PDF map you shared uses only very simple textures and plain colours for the hexes, and that wouldn't need more than a couple of texture bitmap fill options to accomplish, which the standard CC3+ overland styles could likely provide.
2. Once you have your source map as a bitmap image (a simple JPG will be fine), all you need then do is create a new Sheet in your CC3+ drawing, for ease call it "BITMAP", and a new Layer, also "BITMAP", and make sure both layers are active (click in their respective check-boxes, if necessary). Then, using the drop-down menu Draw => Insert file simply navigate to where your source map's image is stored, and click "Open". You'll then be asked to click the "First corner" (in the CC3+ window's command line), so pick a suitable spot in your map, and then enlarge the image to an appropriate size, before clicking to locate the "Second corner". Your map image will then be set-up in your CC3+ map file. Once the image is in-place, you can move, rescale and adjust it by clicking somewhere on the edge of the image (only), when the Command line asks you to "Select entities" for whatever command you want to carry out. You will need to check the map scale is correct once your image is in-place, by using the Info => Distance drop-down menu command to measure between two points on the image whose separation you know (flat to flat of a hex, for example), for simplicity making sure "Snap" is turned off, but "Ortho" is on. If the map image is too large or small for the hex-sizes you want, you can rescale it either by-hand, or (better) numerically, by typing the values into the Command line when prompted (and you can just ask it to multiply or divide with the numerical sizes; you don't need to work that out separately).*
3. Not sure why the hex snap grid isn't working for you, assuming you're trying to duplicate the map whose image you showed, because the terrain is shown as per whole hex there, so it'd be easier for you to just work with drawn hex-shaped polygons (created using the hex-corner points in the snap grid), and then place those hexes where required. As long as the hexes are the correct size for what your final map needs to show, this should be fine, and work with the snap grid.
4. Curves can be tricky, but you can simply draw lines using the straight path (or polygon) option instead. You'll need to click to add more nodes for lines/polygon edges to still seem curved in places, but it avoids the oddness that using the curved drawing tools can create sometimes.
5. You may have the arrow cursor set instead of the crosshairs one (crosshairs show exactly where your cursor is using a crosshair that covers the whole map). To change from one to the other, simply press Ctrl + T (and see this post for another option, as well as a caveat on this command's use).

Without seeing exactly what you're trying to draw, I suspect it'll be hard to give more concrete advice than this, but hopefully this will get you a bit further forward.

* [EDIT: I should also note that when you start to import your map image, you'll be asked what file path to use for the source map's image. For ease, it's preferable to have that source map image in the same folder on your computer as the CC3+ map file, and simply keep the "Store the file's path relative to the current drawing" radio button active - it's the default option for this query panel.]