I am not sure if the following sentence is correct
Nonetheless, I consider that living with and have trained my own dogs have helped me to gain hands-on experience
Can somebody please help me?
The problem with your phrase is that it mixes an infinitive verb form with the simple past, and the lack of sentence construction parallelism makes it difficult to read.
living with and have trained my own dogs
There are many ways of rephrasing this, but you essentially want to make sure that both verbs use the same tense and take the same form. The corrected version that makes the least change is:
living with and having trained my own dogs
As a note, you will also want to make another correction to your sentence:
Nonetheless, I consider that living with and having trained my own dogs has helped me to gain hands-on experience.
Because of how you phrased the sentence, what you're discussing is a single idea that's a combination of the two things.
I consider that X has helped me.
I consider that living with and training my dogs has helped me.
There is a way that you could use have, but the sentence would need to be changed slightly.
I consider that both X and Y have help me.
I consider that both living with and training my dogs have helped me.
You need to decide if it's two things that have individually helped you, or a single collective thing.