英语语法问题The police found （that）the house had beenbroken into a

完整句式应该是：
The police found that the house had been broken into and a lot of things were stolen.
that引导定语从句这里可以省略；
stolen前不加be是因为这里用了过去分词做状语表示状态----东西被偷的状态.
下面有具体用法：
1．过去分词作时间状语相当于一个时间状语从句.例如：
Asked (When he was asked) what had happened,he lowered his head.
当他被问问题的时候,他低下了头.
2．过去分词作原因状语相当于一个原因状语从句.例如：
Frightened (=Because / As she was frightened) by the tiger,the girl didn't dare to sleep alone.
因为害怕老虎,这个女孩不敢单独睡觉.
3．过去分词作条件状语相当于一个条件状语从句.例如：
Grown (If these seeds are grown) in rich soil,these seeds can grow fast.
如果种在肥沃的土壤里,这些种子能长得很快.

that 不可以省略，因为原句形式为find sth and sth，去掉that 引起歧义且有两个动词
完整形式是find sth and find sth done 简化为 find sth and sth done
find sth done 举例： find table cleaned
希望解答你的疑惑

that 后为宾语从句，that 可以省列。　and a lot of things stolen　是一个并列句。句子省列了 had been，以免与前句重复说出，这是并列句常用的省列方法。完整应该是 and a lot of things had been stolen。