Ok as far as paying taxes on LLC income, you are supposed to find as many pre-tax expenses to reduce your income as much as possible with these kinds of entities. (i.e. new computers, server costs, internet fees, etc) If your LLC is making decent profit, the IRS expects you to pay yourself a decent salary for performing work for the company, this money is taxed for FICA, federal, state, and local income tax. Any money you withdrawal as distributions is not subject to FICA taxes, but still the other three. Check with your local CPA for your tax obligations.
From my understanding, the whole point of the thread was to find an entity for the op so he could operate under multiple LLC's while being able to hide his identity from affiliate networks, whether for multiple CPA or adsense accounts or whatever. Op seems to expect no legal trouble via these entities and is using it to mask his identity. In the event of a lawsuit, a court would subpoena the LLC's bank and filing records to find the OP did in fact own/operate the LLC and would be exposed, but this is fine as the OP expects no legal action and is still protected by the LLC in the event it would happen.
This is not a tax avoidance/evasion method, it is just for appearing to these networks to be independent companies operating independently of one another, when in fact they are interlinked via being owned by the OP. Most companies will require a company contact name/number.. you will need to use an alias and be sure to keep the alias straight for each of the companies you operate under in the event that they contact you.
As for bank accounts, all you typically need are the articles of incorporation, and the EIN of the business, as well as a signed company resolution that you are authorized to open and operate bank accounts on behalf of said company. I recently went to Chase to open a bank account for a new LLC, and they looked the company up on the state filing site to check that my name was on the LLC. I'm not sure how this factors in for NM LLC's but I'm sure you'll be able to get an account somewhere.
People looking to offshore corporations/bank accounts in tax haven jurisdictions should note that if you are a US citizen, and performing services on behalf of the corporation in the tax haven, then ALL of the revenue is taxable as your income (no tax deductions allowed.) If one of these companies claims to not share information with foreign governments, and in the future gets on the tax haven blacklist.. and wants off.. they ill comply with the wishes of the US government and the owners of these entities will be open season. I think the usual punishment for filing false returns is 10 years in prison and 250k fine. This is the only reason I've hesitated about going offshore to avoid (not evade) taxes. The IRS is the biggest, most powerful mafia in the world.. anyone that tells you otherwise is delusional.