On using a washing machine for spinning... You need a machine that doesn't add water in the spin cycle; that doesn't tumble / reverse direction as it's getting ready to spin; ideally that lets you stop the spin when you want; and, in my experience, it's best to not spin until you've done at least a couple of rinses, so that there's not much soap left in.
To explain why... felting is a product of agitation, temperature shock (especially hot to cold) and lubricant. So adding cold water to warm, still soapy fleece, is a bad thing. As is tossing it about. Risks much reduced by getting most of the soap out before spinning.
(When I say 'soap', I mean detergent. I use Ecover washing up liquid. You can use Woolite, Power Scour, washing up liquid or shampoo.)
Grease can also be a lubricant, so if you are going to use your machine to spin-dry, make sure you've washed with enough heat (greater than 160F - way too hot for your ungloved hands) and plenty of detergent, so that the grease has been washed out. (Or use Power Scour, which can remove grease at hand-hot.)
If the fleece is greasy, you need to wash very hot with plenty of detergent, not leave it in for more than about 15 minutes, and rinse in water the same temperature (or hotter. Not colder.) If you let it cool with the grease in suspension, the grease can get deposited back on the fleece, and it's harder to remove the second time around.
Very detailed, step-by-step, with pictures, blog posts by Deborah Robson (author of The Fleece and Fibre Sourcebook)
here