Just seen this. I live part of the year in Bucharest (it's complicated...) June is usually hot but not unbearable, could be some rain (think like intense but short-lived summer storms). Therme is great! Be careful about bears, they're actually dangerous but the sanctuary is great. Bran castle is a tourist trap but I have a soft spot for it, you should also go to Sinaia and Brașov while in that area. The city is incredibly safe and welcoming.
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You'd be fine in Bucharest, it's a diverse capital city, people might ask you why you are in the country but they do that to everyone. Outside the city I think you'd be unlucky to counter unfriendliness but you might get stared at!