Horse manure is superb but is best applied when reasonably well rotted i.e. it is dark brown with not too many bits of yellow straw or wood shavings. Usually it reaches this point after at least six months in a heap. Wood shavings take longer to rot than straw but are helped to rot by all the nitrogen in hen droppings: unless you have a huge flock, I would add the chicken coop contents to your compost heaps, which should become bigger once your allotment plot is cropping.
Regarding weed seeds, some discipline is needed when making compost heaps, which rarely heat enough to kill all the seeds, or even half of them. So you need to put less in. My experience is of weeding when weeds are small, before they seed, so my compost is clean. Plants laden with viable seeds are better burnt, sorry not ecological, in my case that amounts to a few foxglove stems, mostly seeding flowers from the herbaceous beds. However I would still use your weedy compost, especially if it is initially under a covering mulch, then be prepared to hoe it later when any seeds have enough daylight to germinate – it may be less than you imagine, but you must keep on top of them.