Log linearising EUler equation
As you say the first step is to take log of both sides after that you are just applying the rules for logarithms and rearrange.

For example:
$$\ln (XZ)=\ln X + \ln Z$$
$$\ln X/Z= \ln X - \ln Z$$
$$\ln X^a = a \ln X$$
$$\ln 1 = 0$$

Also an important approximations that hold close to zero are applied here as well these are:

$\ln(1+x) \approx x $ for $x$ close to zero (which for interest rates and inflation which are usually just couple of percents applies).

Also Taylor approximation is actually a different way how to linearize relationship so although it’s an example of linearization it’s not necessary log-linearization. In fact the result $\ln(1+x)$ is based on Taylor approximation but it’s not log linearization because just applying logs there won’t produce loglinear expression.

Using these rules you can prove all the above solutions. I will leave the first equation for you as an exercise, for the other equations you can see that:

Log linearizing  $1=\beta(1+r)$ gives: $ \ln 1= \ln (\beta(1+r))$ which after simplification gives us $0= \ln \beta + \ln (1+r)$ or $\ln \beta = -r $

From the second equation  $\beta=(1+\rho)^{-1}$ log linearizing gives us $\ln \beta =-\ln(1+\rho) \implies \ln \beta = -\rho$. Hence you get the equality that $-r=\ln \beta = -\rho$ then you can multiply all sides by -1 to move the minus in the middle of the equality.

The $1=\beta (1+r)$ comes from the fact that rational person would want the marginal utility of consumption today and in future to be equal so actually the equation properly reads:

$$u_t^{\prime} = \beta (1+r) u_{t+1}^{\prime}$$

Which can be rewritten as: $u_t^{\prime} / u_{t+1}^{\prime} = \beta (1+r) $ and if $u_{t}^{\prime}= u_{t+1}^{\prime}$ you get the result that $1=\beta (1+r)$. Again this is because in steady state you want marginal utility of consumption to be equal in each and every period.