If I leave the country, will my H-1B application be considered void? Can I leave, get my visa stamped, and re-enter in October?

Yes, if approved you may obtain the H-1B visa at a consulate.

Yes, once the waiver is approved there should be no need to consular process.

Can I ask the landlord to take off some of my final bill because they never fixed some of my property?

You can certainly ask and calmly and politely explain why you are entitled to it.

You can follow the other attorney's advice ASAP.

Am I responsible for the rent on the renewal? The lease states that the start date of that contract is 8/1/16

It will depend on the other provisions contained in the lease.

Shouldn't the 60 days start Monday May 8th? Shouldn't the notice be by way of either certified mail or delivered personally?

As usual, attorney Moss has given excellent advice.

What can I do about these problems I'm having with my landlord?

It seems easiest to move out as soon as your lease is expired.

Eviction based on existing lease with no clause for pets?

Unless the animals are causing unruly disturbances or are restricted breeds

The previous attorney has asked some good questions - instead of contacting the landlord and starting a fight, have you considered simply asking...

Do you know what the landlord is going to try to sue you for (breach of the lease, nonpayment of rent, etc.)?