Normally I don't go for the scruffy, long-haired look, but Mison with short hair looks about a decade younger than Crane
I always go for the scruffy, long haired look- if it suits the person :'). But yes, without the beard he would look younger. And in the 18th century most men of high learning/breeding were clean shaven - I posted something way back that mentioned this. Most people of higher learning/status would not have had a beard.
But in this case, the powers-that-be be decided that Crane would look most out of place with long hair (especially with the leather hair tie) and a beard, compared to the clean-shaven, short-haired people of Sleepy Hollow. But lots of guys have long hair/beards today.
As for the clothes, we only saw in John Doe that Ichabod and Abbie had gone shopping for essentials. Ichabod pulled out a beard trimmer/razor, mirror and brush from the bag. First time. Before that, he would have only the shampoo/soap and whatever is available in the hotel room. I do agree - sometimes his hair looks greasy, but as someone with thin hair, I can tell you that I can wash my hair with substandard shampoo one day, and the next day, it looks greasy. Certains shampoos (and hotel water) do this, so I'll give Crane a break on that.
He just needs some better quality shampoo/conditioner :')
As for cleanliness, we have seen him hand wash his clothes and shower. You are absolutely right - his upbringing would have made him want to appear impeccably groomed and dressed when appearing in public, but he doesn't have that luxury now. Ichabod literally has the clothes on his back, and that's it. And the clothes on his back (coat mostly) are a bit battered up.
Though it's unrealistic for Abbie to have not have maybe nagged him more about clothes, I think SDcat2009 is right,
it's a marker of his recent-for-him past life.
Abbie did comment on it, but maybe he is just being stubborn about replacing clothes. In any of the behind-the scenes photo's of Crane, not in battle, he looks put together and clean. The show is just stalling with the update in clothing.
I wonder how Revolutionary Era cloth would handle a good dry-cleaning. Damn, that coat and boots (and outfit) should technically be in a museum!