Consider this process $$x_t=c+\varphi x_{t-1}+\varepsilon_t$$, with usual assumption of $$E[\varepsilon_t]=0$$
Assuming it's stationary, obtain the mean $$E[x_t]=c+\varphi E[x_{t-1}]$$ Hence, $$E[x_t]=\frac{c}{1-\varphi}$$
This works only when $|\varphi|<1$. If $\varphi=1$ this doesn't work, it blows up, because $E[x_t]\ne E[x_{t-1}]$, i.e. the process is not stationary.
That's what unit root test does, it gets the differences $\Delta x_t$ and regresses them on lagged $x_{t-1}$:$$\Delta x_t=c+(1-\varphi)x_{t-1}+\varepsilon_t$$ Then it tests whether $1-\varphi=0$. If it's zero then we have a unit-root, i.e. non-stationary process.
The constant refers to the term $c$.
The trend is easy too:
$$x_t=c+\alpha t + \varphi x_{t-1}+\varepsilon_t$$
In this case $$E[x_t]=\frac{c+\alpha t}{1-\varphi}$$ So, if you have a trend then the process is non-stationary, but if you account for the time trend, it's still stationary around the line.
Yes H0 is the hypothesis, that you have a unit root in your data.
If you type drift, there will be a constant but no trend in your model as you mentioned.
the critical values are listed at the bottom of the test. So for example for ln(x) the ur.df function gives you the value of test- statistic which in your case is: -2.1074 since this is not smaller than the critical value of -2.86 ( I assume you use the 5% level) you can not reject H0.
For the order of integration: run the same test again, but this time using the first difference of your variable. If you can reject H0 this time your series is I(0) if not you have to take the second difference and then test again. If you can reject H0 after taking the second difference your series is I(2), and so on...
Best Answer
In general, if you decide what hypotheses to test by looking at the data you have to take the resulting p-values with a pinch of salt. The test for stationarity around a trend is the less specific (the slope can be as small as you like), so it's perhaps the better one if you're not prepared to assume beforehand that there's no trend. (And for some tests the null hypothesis is that there is a unit root; for others that there isn't, so you have another choice there.) Many people prefer just to examine the auto-correlation & partial auto-correlation functions for raw, de-trended, & differenced data.
As for the mean of a stationary series: you can certainly test whether it's significantly different from zero. But don't confuse lack of significance with positive evidence that it's exactly equal to zero & therefore remove it just for this reason.