Will the static pressure of a fan change with the RPM & flow rate?

Pease refer to law of affinity

Law 1. With impeller diameter (D) held constant:

Law 1a. Flow is proportional to shaft speed:[1]

Q 1   Q 2 = ( N 1 N 2 ) {\\displaystyle {Q_{1} \\over \\ Q_{2}}={\\left({N_{1} \\over N_{2}}\\right)}}

Law 1b. Pressure or Head is proportional to the square of shaft speed:

H 1 H 2 = ( N 1 N 2 ) 2 {\\displaystyle {H_{1} \\over H_{2}}={\\left({N_{1} \\over N_{2}}\\right)^{2}}}

Law 1c. Power is proportional to the cube of shaft speed:

P 1 P 2 = ( N 1 N 2 ) 3 {\\displaystyle {P_{1} \\over P_{2}}={\\left({N_{1} \\over N_{2}}\\right)^{3}}}

Law 2. With shaft speed (N) held constant: [1]

Law 2a. Flow is proportional to the impeller diameter:

Q 1   Q 2 = ( D 1 D 2 ) {\\displaystyle {Q_{1} \\over \\ Q_{2}}={\\left({D_{1} \\over D_{2}}\\right)}}

Law 2b. Pressure or Head is proportional to the square of the impeller diameter:

H 1 H 2 = ( D 1 D 2 ) 2 {\\displaystyle {H_{1} \\over H_{2}}={\\left({D_{1} \\over D_{2}}\\right)^{2}}}

Law 2c. Power is proportional to the cube of impeller diameter (assuming constant shaft speed):

P 1 P 2 = ( D 1 D 2 ) 3 {\\displaystyle {P_{1} \\over P_{2}}={\\left({D_{1} \\over D_{2}}\\right)^{3}}}

where

• Q {\\displaystyle Q} is the volumetric flow rate (e.g. CFM, GPM or L/s),
• D {\\displaystyle D} is the impeller diameter (e.g. in or mm),
• N {\\displaystyle N} is the shaft rotational speed (e.g. rpm),
• H {\\displaystyle H} is the pressure or head developed by the fan/pump (e.g. psi or Pascal), and
• P {\\displaystyle P} is the shaft power (e.g. W).[2]

So according to the the two laws depending on if you keep diameter size constant  then the ESP will change .

yes the friction losses increase the static pressure 

ESp for a fan will remain same with same rpm and flow rate.

In constant air flow systems the ESP has been calculated for specific flow rate and duct routing what ever it is. During the installation and balancing the system shall be balanced for a specific flow and ESP than for that specific fan the system shall remain under same conditions for ever untill unless the system parameters are same the flow rpm and ESP shall be same.

yes static pressure can be high or low debending on rpm

yes, if fan impeller shaft rpm increase => shaft power increase the air flow increase, the friction losses increase then ESP increase

compared to the incompressible fluid such as water, the friction loss due to the air movement is negligible in straight ducts.

Static pressure again depends on the size of duct, fittings, rpm and flow rate.

the question is not sufficient to justify the "Yes"

No.not affect the only thing affect the type of duct.