Being + Adj Using

Case 1:
If you look carefully at the first part of your sentence, you might have found the answer yourself :)

I *am* sorry for...
Since sorry is an adjective and it plays the role of a complement in a sentence, English requires that you use the verb "to be" in such situations. So, you say: I am sorry; you are sorry; he is sorry.

I *am* sorry that I *am* late
If this is the way how you write your sentence, you will also see that you need the verb "to be" before late because it is also an adjective.

When you switch the sentence to "I am sorry for..." you need to use a noun after for. To make "to be late" a noun phrase, you use the gerund (-ing) form. That is how you get: I am sorry for being late.

If you prefer to use a noun, you can use "lateness." I am sorry for my lateness. In this case, you treat your being late as an event and you apologize for the event. In the case "I am sorry for being late", you apologize for the action.

Case 2:
Similar to what the author above has indicated, "I think being brave is important" and "I think bravery is important" are both grammatically correct. The former means that the action of being brave is important, whereas the latter suggests that the concept or quality is important.

The idea is similar to the differences in:
I like to swim / I like swimming.

When you use a verb, your focus is on the action. When you use a noun, your stress is on the event.

您好，我推薦您一個非常不錯的英語學習網站，里面有很多非常不錯的學習資源，您能從中得到不少幫助：

http://www.hkenglishstudy.info

祝您好運!

試下答你，我唔係英語老師，所以啲grammar理論未必啱。

1.
late係adjective
for之後是交代因由，所以要跟一個noun，如你在2的注解，由於late不是verb，亦只有verb+ing成為gurend可以作為noun的功能，所以用一個最原始的be(是)+ing的字作為名詞再加上所形容的原因。
being late係常用的對話語，可以當口頭禪記住。因為late親都好難自己獨立出現。

例： I am late. 係直接面對人說我遲到；但I am being late.亦正確，但可以顯示誠意的一種說話方式，有點似「我遲到咗添」。

2.
主要原因跟前述一樣，brave是adj.

braveness不是noun.好以冇這個字的。 bravery才是brave的noun.

這句可以正確地寫成：
I think bravery is important.

不過意思有點分別：
I think being brave is important.
我認為做人要勇敢是重要的。
或理解為
我認為做一個勇敢的人是重要的。

I think bravery is important.
我認為勇敢(這質素)是重要的。

希望你看出兩句的分別，這些分別已經不是文法的對錯(兩句都對)，而是表達不同的意思。