The important thing to watch out for grammatical error, among others, is the use of the subject-verb agreement. A singular subject must use a singular verb. A plural subject must use a plural verb. Let's investigate each sentence.
"It is very important to train your dog": 'It' and 'is' are singular. Correct.
"It keeps your dog’s mind active, and it help to build an understanding between you and your dog.": 'It' is singular and the verb 'keeps' is also singular. That is correct. But the other clause of the sentence must have a mistake. It should be 'it helps', not 'it help'.
"Often owners will shout at their dogs or even punish them when what is needed is effective training": 'Owners' is plural so it is correct to pair it with 'shout' and 'punish'.