Permission to learn these 点前 is given together with 小習, but there are more complicated movements that are seen in the 奥伝.

• 利休形 is made of unlacquered paulownia wood, can be small or big.
• Red 一閑張り lacquer ware was favoured by Sōtan.
• Check out this blog making her own amazing 茶箱.

As Burke notes, 利休 possibly just put 茶碗 and 棗 next to each other in the 茶箱, and placed it horizontally in front of the 水指, then lifting them out of it. That makes alot of sense to me.

Now, there are six 点前. Four were originally created by 玄々斎 (Urasenke XI). According to nihonjinron and CQ35, it was in Spring 1854 when he visited 松阪(まつさか) in いせ that he designed the first three (花, 月 and 雪), later adding 卯の花 to complete the seasonal cycle. Then Ennōsai (Urasenke XIII) refined them.

• 雪 (for winter)
• 月 (for autumn)
• 花 (for spring)
• 卯の花 (for summer, but practised all year)

There are two more, which were created by 淡々斎 (XIV):

• 和敬
• 色紙点

• Always after 総礼 the 茶碗 moves!
• KM24 had a great suggestion, even though it isn't canon: You bow at the door with the 茶箱, it's 総礼; but when you bow for the 振出し, you say 「お菓子をどうぞ」. I like this idea a lot.
• Here's a table comparing all 6 点前 with and without 拝見: 茶箱点前

• Because it get's quite long to ask for all the items, KM 24 just say: 「のちほど、拝見を。。。」. This happens after the host has put down the 茶碗 and said 「お仕舞いにいたします」
• If you use a 帛紗 for 拝見, it's best to put it back into your 着物 between the different items, rather than to leave it outside (KM 24). This could be an opportunity to use a 出し帛紗 for 拝見, if you want (KM 26).
• Because often things are a set, either the guest or the host can say that this and that looks similar: 「筅筒と同様(どうよう)でございます」

• It would be ridiculously cute if the inside of the 茶箱 looked like a miniature 茶実, but that's probably childish.
• If I have this fascination with 「海に出て。。。」, the 和敬点 with this poem written on it would be right up my alley.