My bags will fit the sizer, that's not a problem. I have two, that's why I went with the Plus fare (and I need to check in too, it's a long winter trip). But I am not sure they won't be over 10kg especially together. Or even, individually. Do they weigh 'em or just size-check?
In the past ten years I have had my hand luggage weighed only once (and it was not at Stansted, they have never even looked at my hand luggage), its highly unlikely to happen if you check in online and print your own boarding pass, more likely when checking other bags in as you mentioned. If your bag looks bulky or especially heavy they may ask to weigh it, but it really depends who is on the counter at the time.
As per the instructions, your larger cabin bag can weigh 10KG, there is no reported weight limit for the smaller bag you are allowed to take, so you could try to place the heaviest items there, as well as in your 20KG checked luggage. If your bag is only slightly over they are pretty lenient in letting it slide.
So, it's a possibility, but a small one. Try to reduce the weight as much as possible by spreading your items across the bags.
Most of the time, most of the bags will not be weighted. But sometimes some bags are selected and checked.
If your bags are looking small(ish) your chances are better to avoid being selected.
There is a risk that on some flights all bags will be weighted but I hear almost no stories about that happening.