(1)After he had bragged about his way to learn English, he sat down for rest.

(2)=After he had bragged about his way of learning English, he sat dow for a rest.

(3)=After he had bragged about his way of learning English, he sat down to rest.(rest当动词使用，to rest为不定式做目的状语）

(4)=After he had bragged about his way of learning English, he sat down and rested.(rest当动词使用，与前面谓语sat down构成并列谓语）

(5)=Having bragged about his way to learnEnglish, he sat down for rest.(现在分词的完成式作时间状语）

(6)=Having bragged abouthis way of learning English, he sat down for a rest.(现在分词的完成式作时间状语）

(以上5句都是正确的表达）

回答如下：

以上句子是正确的

关于休息一会以下2种表达都是OK 的.但是中间有形容词，不定冠词a不能省略.

(1)for rest

(2)for a rest

(备注：for a good rest 中间冠词不能掉）

因此下面可以互换

sit down for (a) rest=sit down for rest

way of doing 和way to do是可以互换，两者都可以使用

one's way (of) doing sth=one's way( to) do sth

(1) for rest用法

The tired old man craved for rest.

疲倦的老人渴望得到休息

The park on the waterfront is a good place for rest and recreation.

这个滨水公园是休息和娱乐的好地方。

Let my mind dwell for rest and cuddle up.

让我的心灵栖息

It is time for rest and refreshment.

这是时间休息和点心。

A park is usually used for rest and relaxation.

公园通常是用来休息和放松的地方。

He works from dawn to dusk, nor does he pause for rest.

他从早工作到晚，从不停下来休息。

Miss. Li have no time for rest from eight in the morning to nine at night.

从早上八点到晚上九点,李老师都没有时间休息。

(2)for a +形容词+rest(如果rest前面有形容词中间不定冠词a不能省略

You look tired. You need a good rest.
你看来累了。你需要好好休息一下。

(3)for a rest 结构

At intervals she would stop for a rest.

她不时停下来休息。

Stop for a while and take a rest.

停一会儿休息休息。

He set down the heavy box and took a rest for a while.

他放下沉重的箱子歇了一会儿。

The poor old man must be got away from his hard work for a rest.

必须帮助这位可怜的老人摆脱他繁重的工作去休息一下。

(1)  way to do sth

Try to find your own way to express the idea .   

尽量用你自己的语言表达这个意见。

His way to ask me to do something is unacceptable.

他要求我做某事方式令人不能接受的..

What is the best way to clean this?   
清理这个最好的方法是什么?

She showed them the way to do it.   
她向他们示范做这件事的方法。

Try to find your own way to express the idea .   
尽量用你自己的语言来表达这个意见。

What's the right way to say this in English?   
把这句话说成英语,正确的说法是什么?

They are trying to find ways to prevent the disease.   
他们正设法寻找防止这种疾病的方法。

She is in no way to blame.   
根本不应该怪她。

(2)  way of doing sth

They eventually came round to our way of thinking.   

他们终于接受了我们的看法。

His way of sleeping makes me cringe.

他睡觉的方式令我生厌。

We couldn't contact you— we had no way of knowing where you were.   
我们联络不上你——我们无法知道你在哪里。

I wish I knew a way of making money quickly.   
但愿我懂得些快速致富的手段。

He had a strange way of making his classes lively and interesting.   
他有一种奇特的办法,能使他上的课生动有趣。

brag about sth对…自夸，吹嘘…; 夸耀; 自吹自夸;

He's always bragging about his prowess as a cricketer...  

他总是吹嘘自己板球球技的高超。

Don't brag about what you're going to do . Get something done.  

先别吹, 做出具体成绩来再说.

 You just love to brag about things beforehand. "  

你老喜欢吹在前面. ”

 And have to listen to them brag about how popular their daughters were.  

你还得听她们吹嘘自己的女儿有多少人在爱慕.

He loved to talk big and brag about a variety of things to other people.  

他喜欢说大话,很多事情都喜欢跟人乱掰一通.

Who else would brag about meeting Satan halfway?  

谁还会吹嘘他在半路上见到了撒旦?

Let someone else brag about that cool thing you did.  

让别人去吹你做的事有多酷.

It is not good form to brag about winning.  

吹嘘胜利不是好的做派.

 Joe likes to brag about all the friends he has.  

乔伊老是喜欢吹嘘他有好多朋友.

 The University of Florida has two things to brag about this morning.  

今天上午佛罗里达大学有两件事情值得自豪.

For example, you can brag about your site traffic.  

比如你可以吹嘘自己的网站有流量.

 He used to brag about the unique and wonderful fans he sold.  

他常吹嘘他的扇子天下无双、美妙绝伦.

 To brag about oneself.  

这些动词均含有“自夸,吹嘘”之意.

 Untidy work is nothing to brag about.  

不整洁的作业没有什么值得自夸的.

 Average sell records are nothing to Brag about.  

销售成绩平平没有什么值得自夸的.

Would he brag about his conquest?  

他会夸耀他所征服到的爱 吗 ?

(在英语中，有些名词只能接of doing ;有些名词只能接to do ,有些动词of doing 和to do 都可以接的.）