Sentence structure

I'm having a hard time working out the way sentences are formed in English. I have found in grammar books the following statements: "The simple sentence". A simple sentence consists of just one independent clause ;No matter how long it might be. For instance: "I love dogs" and "The badly dressed woman standing right in front of you is my exgirlfriend" are both simple sentences??

2- compound sentence. The compound sentence consists of two Independent clauses joined by an coordinating conjunction. For instance : "I like swimming,and my sister likes surfing" and " I want to go to the party,but i have no money" are both compound sentences?

First of all, are the concepts mentioned above correct?

Secondly, should both clauses in a compound sentence "always" be independent?

If i wrote " She plays volley very well,but i play better" what type of sentence would this be?

Mastering this aspect is fundamental to me because i need to learn English punctuation correctly.

All sources of information are highly appreciated.