The client sent me a letter asking if we could change the information for them.
This sentence is OK grammatically. But mind that in most contexts it would mean:
The client sent me a letter. The letter contained his request to change some information for the client.
The second sentence is also grammatical:
The client sent me a letter and asked if we could change the information for him.
But note the difference in meaning:
The client sent me a letter. He also asked (by phone, or personally, or in other way, but not in the letter) if we could change some information for him.
To make the second sentence identical in meaning to the first, use the relative pronoun that:
The client sent me a letter that asked if we could change the information for him.
If you want to investigate the grammar behind all this, read up on "relative clauses" (they use relative pronouns like that) and "participial clauses" (they use verbs ending in -ing and -ed).
You ask:
I'm confused with the first one, why past tense can be used with verb+ing?
Because the verb+ing (present participle) really connects not to the verb sent but to the noun letter:
- What kind of letter? - Letter asking if we could change some information.