That very much depends on a lot of different factors.
In wild sheep or domesticated self-moulting sheep, the fleece may just lie there and decompose, unless taken by another animal or bird for burrows or nests.
If the sheep is a meat sheep, the fleece is usually of very low quality. Occasionally, it can be sold as insulation material or for industrial carpeting, but it is quite often just dumped and allowed to decompose.
Better-quality fleece can be made into high-quality carpeting, upholstery and soft furnishings; clothing; or yarn for fibre arts – knitting, weaving, etc. In some countries, there is a central processing board to which all such fleece must be sold (e.g., British Wool, in the UK), in others there may be competing processers to which farmers can sell fleece.
No. It’s like you getting a haircut.
Trimming is the same as cutting, in hair terms – except less hair is removed. Trimming wool does not help the sheep; it leaves most of the wool in place which is uncomfortable and unhealthy. In addition, the trimmed wool ends would be too short to have any commercial value and would end up in landfill. As wool flocks would no longer have any commercial value, thousands of sheep farmers would go out of business and their flocks would be slaughtered. Those industries dependent on sheep wool would have to turn instead to synthetic alternatives which are very damaging to the environment.
The main issue with storing wool is making certain that moths don’t get at it. Wool can survive almost anything else.
The best method is sealing it in something airtight. This might be some kind of sealable plastic box, or vacuum bags – the kind that you suck the air out of with your vacuum cleaner. NB: the vacuum method may leave the wool flat and crinkled, but you can restore it by steaming it for a few hours. Place the wool in your freezer for a few days before storage: this should kill off any moth eggs that might inadvertently have got into it (this also works for wool garments that have been infested with moth larva).
However, if you mean raw wool, just off the sheep, don’t seal it in plastic: that will just allow it to stew in the sweat and grease. Might not harm the wool, but no one will want to touch it afterwards. Raw wool should be scoured – cleaned – before long-term storage, and placed in an air-permeable bag – hessian, cotton, or hemp is good, or failing that an open plastic bag – if you’re just taking it to be scoured.