Questions on comparing GDP between various countries
Answer to question 1:

OER is an actual price set by the government. The real exchange rate or RER (That´s how people typically call it) is a ratio of prices of goods, as you mention. 

The arbitrage does not go away because the goods in the RER calculation are not completely tradable. The PPP among tradable goods is closer to 1 (meaning the PPP exchange rate among tradable goods is closer to the OER). You can´t trade hamburgers, they get cold, mushy....

Moreover, goods and services are not the only thing being traded across countries. You also trade capital. So a country that is in vogue for international investors will see lots of demand for its capital, raising the price of everything denominated in its exchange rate. So its goods and services look expensive, but its not that things are in disequilibrium its just a disequilibrium in the goods and services part of the balance of payments, that is balanced by another disequilibrium in the capital part of the balance of payments.

Answer to question 2:

Use nominal GDP, in Dollars or in PPP Dollars. Use dollars if you care about dollars, for example if you want to figure out the likely market size for your good which is produced in the US. Use PPP dollars if you care about local living conditions. PPP adjusted (per capita) GDP is a good way to figure out (economic) living conditions.

Answer to question 3:

High rates should imply higher returns to investors that provide liquidity to the local economy, so they should be connected with a more appreciated currency. High inflation lowers the expected future value of the currency, so it should make holding the currency less desirable instead. This should lead to a more depreciated currency.

However, the answer quickly gets more complicated if you add expectations and monetary policy to the mix. For example, an increase in interest rates that will lead to an expectation of a slowdown in the economy and very low future rates might depreciate the currency; high inflation that will lead to an expectation of a big monetary policy contraction raising interest rates, might appreciate the currency.