I'm really confused which thickness of rubber sheet I should choose. Does the thickness affect the tensile strength and elongation in any way? Is there any relationship between them?
The thickness of your rubber sheet definitely affects the force you will need to apply for a given amount of elongation. Thickness also affects the breaking point, and therefore the tensile strength, of your rubber sheet.
Young's modulus describes the elastic property of a material up to the limit of its elasticity. It's a measure of the force per cross sectional unit area required to stretch or compress a material. It's the ratio of tensile stress (force per cross sectional unit area) to extensional strain (ratio of the amount of deformation to the initial length), along an axis subject to Hooke's law.
Young's modulus is expressed as pressure. For rubber, it's 0.01 to 0.10 Gpa (per the table in this link: http://www.engineeringtoolbox.com/young-modulus-d_417.html). If you click on the hyperlink in the paragraph above and scroll down to "Calculation", you can see how the modulus is calculated. You can use this calculation with the specific modulus for rubber to determine the thickness of sheet you want for a particular force and elongation.
Strength is related to elasticity, but it describes the deformation of a material past the limit of elasticity. The yield strength is the point at which stress applied to a material begins to cause it to lose its elasticity - to lose the ability to return to its original shape. This is the point where plastic deformation begins, as opposed to elastic deformation. Ultimate tensile strength measures the maximum stress that can be applied to a stretched material without the material failing.
Yield strength and ultimate strength are shown in stress-strain curves that plot force per cross-sectional unit area against the ratio of change in length to original length.