When it comes to English, my accent is probably a mess. I've never really tried to emulate any kind of accent but I guess I've always favoured General American more than Received Pronunciation. It somehow felt more natural for me to speak, so I've stuck with it, although I have some occasional slips of RP and who knows what else. Throw in a bit of Eastern European accent, mix it all up with GA and RP and...nobody can figure out where I'm from.
I love accents and since English has been my point of interest for so long, I'm able to distinguish and enjoy the different varieties. It's like a hobby. So, I don't think I really have any stereotypical notions about different accents.
It definitely takes some time to get used to people speaking with a thick accent. But it also depends on the accent, some are easier to get used to than others. I used to have a lecturer who had a pretty thick Scottish accent and it took some adjustment at first. I never had any problems with some others.
I've noticed that I do modify my accent completely unintentionally. But it depends on the people I'm speaking with, it's like I'm supporting them to make the atmosphere more comfortable, sort of mirroring them. If they have a distinguishable accent that is different from mine (whatever nationality, speaking English) I start taking on their accent unintentionally. Also, when rewinding the conversations in my mind later on, or even thinking about the person, I notice my inner voice using the same accent (I mostly think in English anyway).