I know the LHC people are publishing new limits every now and then, but as a non-expert in reading experimental papers (yet), I was wondering if there's a friendly website that collects and presents the limits in a readable and accessible way for non-experts?
There isn't any such website. Aside from the arxiv, I usually look for new experimental results posted as publicly-available notes on the web, e.g. here from CMS and here from ATLAS. But those are the experimental papers, not a digested form like you're asking about. They have gotten much better about presenting material in a useful way, though; for instance a recent ATLAS result includes this plot on limits for a gluino decaying to a neutralino and a top/antitop pair. Last year, many of us in the theory community were spending a great deal of time simulating and re-interpreting experimental results to understand the limits on scenarios like this. Now they're usually right there in the paper.
I think the Higgs boson mass is by far the single most important constraint on supersymmetry to come from the LHC so far, although the direct searches matter too. The new possible gamma-ray line signal is another interesting development, and other new results from outside the LHC, like an update on the electron EDM, are going to come our way this year too. So there are a lot of developments to keep an eye on. Honestly, aside from monitoring the arxiv and the experimental websites, I think the best public source for "readable and accessible" updates may be Adam Falkowski's Resonaances blog.