Is the reference voltage \$V_{ref} = 0.625\$V?
We can't say that. And it's not indicated in the datasheet. The reference voltage determines the full scale voltage but this doesn't necessarily mean that the reference voltage equals to the full scale voltage (e.g. The ADC might be doing some mathematical operations inside). All we know is that the full-scale voltage is 0.625V. The reference could be something else.
Then could I say, as the AD9361 has 12-bit ADC, LSB = \$\frac{V_{ref}}{2^{11}}=0.305\$mV?
Your LSB calculation is wrong. LSB is ~153 μV:
$$
\text{LSB}=\frac{V_{FS}}{2^N-1}=\frac{0.625}{2^{12}-1}=152.6 \ \mathrm{\mu V}
$$
where \$V_{FS}\$ is the full scale voltage, and \$N\$ is the number of bits that the ADC generates.
Here I don't understand how we get the 63.5 dBFS because \$20 \log_{10}(2^{11}) = 66.22\$dB.
It doesn't seem like you can directly relate the numbers here.
"Low power threshold" is a programmable number given in -dBFS (minus decibels relative to full scale, which basically means less than 0.625V). In this case, anything down to -63.5 dBFS can be selected as a low power threshold (If you read the datasheet further, some functions like "auto incremental gain" can be used in case the incoming signal level is lower than this threshold).
So, -63.5 dBFS corresponds to ~418 μV:
$$
-63.5 = 20 \ \log\frac{V_{LP}}{V_{FS}}=20 \ \log\frac{V_{LP}}{0.625} \\
\Rightarrow V_{LP}=417.7 \ \mathrm{\mu V}
$$
The "resolution 0.5 dBFS per LSB" also makes me clueless.
This is a multiplicator that you use when programming the low power threshold value. If the resolution was 1 dBFS / LSB then writing 0x15 to the settings register (assuming an 8-bit register) would set the threshold value to -21 dBFS. Since it's 0.5 dBFS / LSB, writing 0x15 will set the threshold to -10.5 dBFS. To program a low power threshold of -63.5 dBFS you'll need to write 127 or 0x7F.
