Schema / Rich markup doesn't necessary mean your rankings will improve, it similar for better markup and most things Google will learn itself without the additional markup.
Looking at the link you provided this looks similar to gumtree which is personal ads, which would fall under the shop category since this is what Google defines board for anything that is a sell/swap/auction.
Anyhow, thin content is extremely different to rank and even a 1,000 links wouldn't overcome this problem on some of the pages, Gumtree, Craiglists which also has a fair bit of thin content have millions of links.
Looking at the frontpage there's room for improvement such as beefing it up with what it is your website is all about you can use this in slide out boxes or anything with display:none that is 'triggable' if you want to keep your pages minimalistic. You can also add things like privacy policies, terms and conditions to lower your content ratio across the site as a whole.. I'm sure that Google will do a check on X pages and how many words / by but of course no one truely knows but this would be logic thinking here.
So as you already know the main problem is the thin content, so with less you need more... Simple having a few links from some Good websites won't go unnoticed but its hardly going make your site rock due to low content value will be giving it...
It's all very well saying the site is a listing site, its bound to have thin content well this isn't Google stances and you should be looking at ways of giving your users more information.
For example take https://trashswag.com/reports/view/987#.UnRfyBDLJ8E as a prime example: simply having 1 picture and Internal door. Veneered in text is not going to help the site in any shape of form and the more pages like this you have the more damaging as a whole it can have on the site.
Motivate users to say a little bit more....
You need to either motivate yourself to write more or motivate the users adding the content, if users are creating the content then you need to make things easy on the people adding the content, eBay is a prime example they have templates for everything being added so in this case people can easily select things like the color of the door, the dimensions of the door, the lock system the door is compatible and so forth.
So ideally you'd have a system that ask the user what type of item they wanted to list and in the same example if they clicked door they could then use drop down boxes giving it additional information such as the size, color as mentioned.
Google can't mind read what pages about
Also looking at most of these pages its impossible to know what each item is without looking at the picture, and if I can only tell by looking at the picture - Google certainly won't - your problem is content and that's not going to be solved by simple telling a instruction even if did one exist.
The last option would I personally wouldn't go for but if you don't care about people's listing be scanned by Google due to short life the other option would be to not index it and just have your site being found.