#1 you don't have to mention the place (context can help you know where without saying it). #2 as you can see in sentence #3, no it can go before or in between. Korean has flexible word order, though sometimes, of course, there is a set way. But sometimes it doesn't matter or changing the word order can alter the meaning/emphasis. Here, you can do 에 (으)러 가다 or (으)러 에 가다. #3 Yes. The object goes with the verb. So of course you aren't just limited to saying "buy" or "eat". You can say WHAT you buy or eat. I went to the store to buy a new bike. I went to the restaurant to eat lunch. I went downstairs to turn off the fan.