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For a constant velocity of the charge, does $t_r=t$? No. Consider a charge moving in the positive x direction at speed $v > 0$, and assume for simplicity that at $t=0$, the particle is at $x=y=z=0$ Then at $t=0$ for a point at $x=d > 0$ the retarded time $t_r$ is at a time on the past light cone. For the charge $x(t)=vt$, so we want a retarded time ...

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An Electric Field is only conservative if it is static. The propagation of E with a L-W field contradicts this, so it is not conservative.

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Your thought experiment stumbles upon an important idea in electrodynamics which is quite counter-intuitive.The EM field produced as radiation due to the charge in fact produces a reaction force on the charge itself. This is known as the Abraham–Lorentz force which is proportional to rate of change of acceleration of the charge. In SI units it is given by, ...

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