1. You can get your visa stamped in any country which is called Third Country National (TCN) processing. It is a good idea to check with them beforehand. Sometimes if they have too much of a workload, especially in Mexico they might temporarily stop taking TCN's.
2. I don't know because I don't know the rules for the drop box. If you read the rules and you qualify for them, then the answer is yes.
3. I-94 is important. Once you are inside the USA your status is governed by the I-94. You could have a visa good till 2020 but if your I-94 is expiring tomorrow then you are going to be out of status day after tomorrow and unlawfully present. If your I-94 is expiring you got to get that extended whether it is through an H-1 extension or if you already have an approval and you can go outside with the visa for a visa stamping. H-1 and L-1 visas require a prior approval from USCIS. H-4 and L-2 visas do not. More...
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