He sitting at the back of the classroom and his eyes staring at the blackboard.是错误的.
前后两部分全是ing形式的现在分词,是非谓语动词,所以整个句子缺少谓语.
Sitting at the back of the classroom,his eyes stared at the blackboard.
这个就完全正确.
Sitting at the back of the classroom是现在分词做状语
后面的his eyes stared at the blackboard已经是一个完整的句子了,主谓宾俱全.
可以写成这样,也是正确的：He is sitting at the back of the classroom,his eyes staring at the blackboard.
这就变成了He is sitting at the back of the classroom是一个完整的句子
his eyes staring at the blackboard是现在分词做状语
或者直接用and连接两个并列句,也是正确的：
He is sitting at the back of the classroom and his eyes are staring at the blackboard.
这样的话,and前后都是完整的句子.