@Cratos I'm going to try and answer as many points as possible! Disclaimer: I work in Maths Outreach for Cambridge - in particular I run the STEP Support Programme (maths.org/step). I also interview and supervise first year undergraduates.
Technically anyone can enter STEP, so you can enter in year 12 but we don't advise it. Most candidates will not have covered the specification by then, and by the nature of STEP questions it can be quite hard to know when you look at them which specific areas of maths they will need so it would be hard to select questions based on just one year of content. By the end of year 13 A level candidates should have covered all of the Pure content and, depending on module choice, either the stats or mechanics and probably quite a lot of both.
If you end up sitting STEP and getting grades S,S in year 12 then you would probably get an offer just based on A levels BUT this is very hard! Anything less than S,S (an especially less than 1,1) and you would probably have to sit it again in year 13 (which is quite expensive).
Maths at Cambridge requires STEP 2 and STEP 3. Other universities which accept STEP usually do so as part of a range of offers (for example Warwick will accept STEP or TMUA, and either STEP paper will do: https://warwick.ac.uk/fac/sci/maths/studywithus/ug/our-offer/).
Oxford and Cambridge admissions work differently. Cambridge interview a high proportion of applicants (80%+ and the proportion is highest for UK state school students), make offers to about 500 students and then uses STEP to make the final decisions down to 250. Colleges tend to make offers to those that they think have the potential to make the STEP grades in 6 months time, even if they might not perform quite as well at interview. Oxford use the MAT to select which students to interview and then the number of offers is pretty much the same as the number of places.
The Cambridge and Oxford maths courses do differ a bit (Oxford offer some "maths and" courses, whereas Cambridge has a lot of theoretical physics options), so it's important to consider this as well as the difference in admissions processes!
TMUA (and MAT) are more-or-less based on half of a single maths A-level (as that is what everyone will have covered) with a couple of extra topics. STEP is based on A level maths and further maths. TMUA is needed for Economics at Cambridge.
Since your DS is at the start of year 12 then there is loads of time to start doing stuff! I would suggest starting on the STEP Support Programme foundation assignments (https://maths.org/step/assignments) which will be useful for any maths heavy university course. It's also a gentle-ish introduction to STEP. There is also the Oxford Online Maths Club (https://www.maths.ox.ac.uk/outreach/oxford-online-maths-club) which runs Jan-June of year 12 (followed by MAT prep June - November, this is also really good TMUA prep).
For super-curricular ideas see: https://www.undergraduate.study.cam.ac.uk/files/publications/super-curricular_suggestions.pdf.
Hope this helps!