For the first set of sentences, they are both fairly equal. In my experience, the 'While' option is more common, but I suspect that this might be a regional thing more than anything else. For the second set of sentences, the 'While' option is definitely more natural without further details being provided. However, I would say that for the second clause, you might choose to say something like 'I plan to visit a lot of countries' or 'I'd like to visit as many countries as I can'. The 'going to' structure implies that you've already made plans so it's common to include details about the plans to reinforce your use of this structure. For example, 'When I've got my Christmas study break during my PhD, I'm going to visit France and Germany to visit friends I met during my masters.'