Do I need to issue a 1099 to a company who I paid for rental services of equipment?

Yes, per the IRS, Box 1 of a 1099 MISC form should include payments made for the rental of equipment.

Here is more information from the 1099 MISC form instructions:  

Box 1. Rents:   Enter amounts of $600 or more for all types of rents, such as any of the following. 

• Real estate rentals paid for office space. However, you do not have to report these payments on Form 1099-MISC if you paid them to a real estate agent or property manager. But the real estate agent or property manager must use Form 1099-MISC to report the rent paid over to the property owner. See Regulations sections 1.6041-3(d) and 1.6041-1(e)(5), Example 5. 

• Machine rentals (for example, renting a bulldozer to level your parking lot). If the machine rental is part of a contract that includes both the use of the machine and the operator, prorate the rental between the rent of the machine (report that in box 1) and the operator's charge (report that as NEC in box 7). 

• Pasture rentals (for example, farmers paying for the use of grazing land).

While the IRS does require you to issue a 1099-Misc form when you rent equipment, the IRS also supplies us with a list of exceptions. You are not required to file a 1099-Misc form if any of the following situations apply:

• You are not engaged in a trade or business.
• You are engaged in a trade or business and
  • the payment was made to another business that is incorporated, but was not for medical or legal services or
  • the sum of all payments made to the person or unincorporated business is less than $600 in one tax year.
  • For additional information, see the following IRS article: Am I Required to File a Form 1099 or Other Information Return?

If you rented the copier from the manufacturer or another incorporated business, you will generally meet the "business is incorporated" exception.

If I provide uniforms to employees do I need a 1099 from the uniform company that I rent the uniforms from?

In it's simplest form, you are not required to issue a 1099 to a *CORPORATION* regardless of the amount. However, you are required to issue a 1099 to an *INDIVIDUAL* you paid more than $600 to during the tax year.

In most cases, an individual includes an LLC since an LLC is considered a disregarded entity for tax purposes.

If I have a small business from my own. I paid rent for a storage under my name. Can I file 1099 MISC using my name and claim as an expense?