I've not tried that brand, but do use my close caboo (DD now 12mo), which sounds similar (fabric ring sling). When I put it on, I make sure there is about 10cm between me and the fabric if I put my hands between it and stretch (does that make sense?), and then slip DD in, making sure her legs are in the frog position etc.
Make sure her face is never covered by the fabric, and she shouldn't need her snow suit on in the sling, just a cot or thin fleece and a hat and booties, if her feet are out. Your body temp will rise a bit with the exertion of wearing your baby out and about, and you don't want baby to overhear.
Have a look on YouTube of people using your brand of sling, and you should get some tips.
Is there a sling meet you could go to for first hand advice? It's easy for them to overheat in a sling because of the warmth of your body so I'd not use the snow suit outside - go for layers, ideally ones that provide extra warmth for bits outside the sling (warm socks, hat etc) and which you can take on and off easily to adjust temperature.
YY to getting rid of the snowsuit. Your should always dress a baby more lightly in a sling than they would be dressed if they were in a pram - your body heat is the equivalent of a snowsuit already!
Head position in a tiny baby is often with face against chest, at your upper breastbone. Most babies breathe just fine this way as their flattened noses let air in on the side. Other babies turn their head sideways, which is also fine. Just make sure the head isn't nodding forward - straight or tipped back a little is best.
Babies are usually fairly snug in a sling btw - I was surprised at first by how snug! - and they're safer when held securely. A sling meet is great for getting help with this aspect.