Do I need to test for assumptions before linear regression if I get good predictive results? Do I need to test for assumptions before linear regression if I get good predictive results? Does the good results imply the assumptions are satisfied?

No, you can have strong predictive performance yet still violate assumptions. Let's do an R simulation where the usual assumptions are violated, yet predictive performance, measured with $R^2$, $R^2_{adj}$, and $MAE$, is strong.
set.seed(2021)
N <- 1000
a <- -2
b <- 2
x <- seq(a, b, (b - a)/(N - 1))
y <- 10*x + x^3 + rnorm(N, 0, 0.1)
plot(x, y)
L <- lm(y ~ x) 
summary(L)$r.squared
summary(L)$adj.r.squared
mean(abs(predict(L) - y))

The regression $R^2 = 0.992864463374598$, $R^2_{adj} = 0.9928573135383$, and $MAE = 1.0491595205924$. The predictive performance appears to be pretty good; we explain most of the variability and tend to miss the true value by only about $1$ unit. However, there is a bad violation of assumptions. Look at the residual plot via plot(L).

