I don't think the Cybex is a huge amount bigger than the Pixel, and they will both be fine to go straight into Axkid Minikid, even if you have the older one. The newer one is suitable from about 4-6 months so there's no worries with any infant seat with that any more, you can even go for the carrycot style if you want to (but they might be less safe.)
The Cosmo is just a more basic version whereas Pixel Pro is more premium. It's common for companies to have two (or more) versions of a seat like this. Safety wise they look the same, the Cosmo even has a slightly better safety rating (the difference is negligible).
The inserts are different - the Cosmo is a one-piece insert which comes out when the baby is over a certain size (which is standard for most infant car seats) whereas the Pixel has a more adjustable insert which will give you a much more tailored fit at every age and stage. I'd say if you're a car seat or orthopaedics nut (said affectionately!) who loves tinkering and is really pleased with getting an exact perfect fit at all times, then the Pixel will make you very very happy in this aspect. It has even won awards for back support. I haven't seen the Cosmo in person, but I have seen the Pixel in person and (as a car seat nut myself) the fit is extremely good. I have never seen a better fitting newborn. Newborns do not often get a good fit in any car seat, so it's surprising when they do.
But if you are a more normal person who would not geek out over these things, and in fact the idea of an insert with a million parts which can all be taken out in different orders brings you out in a cold sweat at the thought of getting it wrong, and you only really care that your baby is safe and comfortable - you'll probably be perfectly happy with the Cosmo or, indeed, the Britax seat. They both give a lovely fit for newborns (that I've seen from photos) and are simple to use and understand. The fact that you can't adjust it for every single 1lb of growth does not really matter.
In terms of the base - with most car seats as sold in the UK, you can fit it by using the car's seatbelt to fit the seat directly into the car, which is easy to do and takes about 30-60 seconds with practice, but if you prefer, you can buy a separate base and fit the base (semi) permanently into the car. The benefit of the base is that you just press a button and you can lift the seat out without having to unstrap a seatbelt, so it's faster, easier and in theory it's less prone to error because all you have to do is push the seat into the base and listen for a click, some also have a little green indicator to show it's in the base properly.
Most bases sold today fit with isofix, so if you want to use your infant seat with isofix, most of the time you'll need a base. (Nuna Urbn is the exception). You can also get seatbelt fitted bases. As said, the benefit of the base is that it stays installed all the time so you aren't having to strap the seat in and out.
Some bases spin, but IMO this is not a useful feature for an infant seat except in a few niche scenarios.
You can get bases which take the next stage seat too but this is usually a bit of a marketing trick - it's very very rarely worth it.
In America they include the base with the car seat which I always think makes a lot more sense!