1) Is it possible to get cURL enabled? Or do you suggest any other method to connect to an external API?
No, that will not be possible for security reasons. You could do some really nasty things with cURL.
However, If you use a local server, you can simply allow that function.
There is a function readAPI() that does currently not support many APIs, but we can extend that on demand.
The question is: Do you really need to adjust the questionnaire on a "live" base while it is online? Or would it be sufficient to get the information from the database in advance?
The databank feature would allow for that, although it's possibly not a good idea to import 136k entries there.
I strongly doubt that there is a research design that will benefit from having such a broad database. Would you mind to explain your idea a bit more in detail?
3) The igdb.com database contains 136.000 records. If I can get this data, may I use "Options from Database" method for "Suggesting Text Input" on SosciSurvey?
Sure. But, again ... this will not be very useful unless you have 500.000 respondents. You will probably not be able to work with records that rate (or whatever) a game, if no other record for that game is collected. So, assume you expect 500 respondents and a minimum of 5 records per game required. Then it will be sufficient to include games that have a market share of 1% within your target group. Maybe go down to 0.5% for safety, but that will still strongly reduce the dataset by one or two magnitudes.