Start up with multiple foreign and domestic partners
How the business and the individual owners are taxed will depend on how you structure the business. If the business is a partnership, then any profit or loss from the business will be considered US source income and the US and non-US owners will need to file a US tax return and report their proportionate share of profit or loss from the partnership.
If the business is structured as a corporation, then the business itself will file a tax return and pay taxes, but generally the non-US owners will not need to file a US tax return, unless they have other US sourced income. If the non-US owners are paid to provide services to the business and are performing those services in their home country, then generally that income would also be taxed in their home country and any dividends they may receive from the corporation would be taxed according to where the individual owners live. Payments by a corporation for services rendered by owners are generally tax deductible, although they may have to be reported depending on what percentage ownership the individual holds. Dividends paid by a corporation are typically not deductible in calculating the corporate tax, but are still taxable income to the owners.
The UK, Czech Republic, Switzerland and Ireland all have a tax treaty with the US, so as a general rule the non-US owners should not have to pay tax twice to different countries, either as a partnership or a corporation.
There are a number of factors that determine where the business needs to be located. Physical location of business operations, any business assets and any employees may require you to register the business in a particular state. If none of those issues impact the business, then of Illinois, Ohio and Texas you would probably want to choose Texas, because it has no state income tax and thus the owners living in other states or not in the US would not have to worry about state level income tax, except for their local state of residence.