Proof that @Belthazar has at least four thumbs…
conspiracy confirmed…
Oh, I’m pretty sure all of my fingers have appeared in these photos at least once each. 
You heard it here first! Suspect accepts charges!
28-down イオウ
which reinforces my suspicion that 29-across is オワビ
Which means 33-down is ワカ
Thinking perhaps 19-across is ミズヨウカン
I’d say it ends with ン anyway, because I’m quite certain that 35-down is something-ビン. Maybe just as simple as ユウビン?
And 23-down is チーズ
14-down ユドウフ (I had that in Kifune 
)
15-across ウルチマイ (Honestly had to look that one up - I’m not well-informed on all the rice terminology aside from 稲 > 米 > ご飯)
I‘m also pretty sure it‘s ユウビン.
And wow, learning more possible words for rice 
Ok, so this time I’ll try to restrain myself and only do a few at a time. 
13 across ウサバラシ
12 down イドウ?
30 across ノシ
1 down カナモノ
9 down ミカン
25 across ムネアテ
Ah, of course. This is one of those ones where I could basically see the expected answer, but my Japanese vocab is not strong enough to come up with an answer.
Which makes 24-down ラム, which is a crossword answer that’d literally only work in Japanese.
I’m not sure about the rest of the word, though. Think 27-down has to be トシナミ
How about ムナモト
More findings:
2-across ナンド (thus 8-down is イン)
27-across トウヨ
34-down ショートヘア, but I’m not too clear on why the clue is negative
31-across ドア? Seems too easy.
32-across … マユ?
31-down … … … ドラマ?
Are we gonna alternate taking two hours to solve a crossword with taking two months?
Yea, I thought it was asking what male hair usually isn’t but maybe I’m misinterpreting it.
A couple more long shot possibilities:
10 across カイフク?
26 down オオダマ?
Well, give me a couple more days to use the answers you just posted to figure out some others and then we can call it quits. I don’t want to get too carried away by this thing like the first time.
22-across: キノジ (so it works with 28-down)
14-down: メシ?
15-down: マワシ
28-down: タカラクジ？
30-down: ノウ
I made too many changes (changing the picture’s size), so I was locked out of editing. 
Apparently it’s that the images are too big (or something). 
Also, in general. Me, while answering the easiest ones:
Edit #42:
Apparently adding images by using Insert > Image > By url, adds the images in a reasonable size.
I was reading Haruhi, and her hair is described as ロングヘア. With that in mind, I’m now starting to think the meaning of this clue is because ショートヘア is explicitly a women’s hairstyle. Men’s hair is expected to be short, so it doesn’t have a name. And indeed, doing a Google image search for ショートヘア invariably shows images of women (or, at least, I think it’s invariably - one or two are a little hard to tell).
If your brain is the thickest part of your body, you’re doing something wrong.
It’s ドウ
Which makes 7-across ミキ
6-down サミット
5-across バクハツ
31-across ウシロ
I went with fattest, as in most fat, not thickest. According to some internet searches.
For 1-down, ひさかたの appears to be a poem of some description - possibly one of the 百人一首. I assume ももしきの is also, but I don’t know what to do with that.
13-down イロハウタ. Duh.
4-across トロロ
17-down ツタ
11-across オウタイ
21-down イキドマリ？
8-across トリシマリ？？
1 down マクラコトバ
1 across マグレ？
9 down グンテ
19 across フォアグラ
34 down ランカン
32 across リカ？
edit: 12 down レイ？
edit: 14 across メザマシドケイ
33 down コケシ
Why? Like, what do the words in the clue mean?
Think you’re over-thinking this one - グラブ works just fine, and has the added advantage of making 3-across ラブ
2-across クライマックス
(which makes 27-down アス)
29-across クルマ? Because there aren’t any trains?
32-down リサイクル
(which makes 28-across タサン)
25-across ガラクタ
35-down タマムシイロ (I don’t really get it, but it fits)
(which makes 33-across コシ)
Words in the clue didn’t mean much. The answer was in the page you referenced.
ひさかたの 日・空など天体に関する語にかかる枕詞（まくらことば）
edit: 16 across ワカイ
19 down フツカヨイ