I'm currently doing my bachelor's degree in mathematics in Germany (Heidelberg to be precise) and I'll soon be applying for master programs / grad school. I basically consider two options:
I) First do a master's degree and then a PhD in Europe and try to get a postdoc position in the US afterwards
II) Go to grad school in the US now and then look for a postdoc position in the US
Concerning I) I really like how math programs at universities in Germany (and other European countries) are structured and if I decide to stay in Europe for now, I will probably do my master's degree at Bonn. However, I am afraid that I'll have a hard time finding a postdoc position in the US having done my PhD in Europe, both because my experience (for instance regarding teaching) might not be what US universities are looking for and because European universities might not have as good of a reputation as universities in the US. I suspect, at least a PhD from Oxford or Cambridge might have some prestige in the US, however, I really don't like the 9 month master system there, as I feel like the material is rushed through and as I've heard, in order to get admitted to a PhD there, it is strongly advised to also do your master's there.
Concerning II) Since I prefer the European master/PhD system, for me to be happy with moving to the US now, it would be helpful to get a position at a top university. I already (successfully) did the TOEFL test but I would have to do the GRE (both general and math) test. Together with application fees this would probably add up to about 1000$ which is a hell of lot of money.
So I guess my questions are:
1) Are my concerns regarding I) justified? How hard is it for someone who just completed their PhD in Europe to find a job at a university in the US?
2) What is the reputation of places like Heidelberg, Bonn, ETH Zürich,... in the US? Is a PhD from Oxbridge prestigious enough for it to be worth it to overcome my aversion towards their program?
3) Do I even stand a chance of getting admitted to a top US program? I took a lot of courses and managed to score between 1.0 and 1.3 (if I looked it up correctly, this should be between 3.7 and 4.0 GPA?) almost everywhere. However I haven't participated in any REUs and don't have anything to offer besides the courses I took. Having read what people that got admitted to Harvard, Princeton, Berkely... have done prior to grad school, I don't think I can keep up with them.
4) Having said that, do you think it is even worth applying and spending such a huge amount of money in the application process?
5) Do you have any other piece of advice or experience to share?
Thank you so much for your help!