Yes, you should mimic a British accent. Your first impression should be correct choice for you. There are different types of accents and some would be considered too strong to be understood, but you probably won't gravitate to those. I recommend you watch all episodes of Peep Show on Hulu. The characters are English with a few Scots. The shots are mostly close ups so you can really study the mouth movements. The characters are very affected and therefore easy to mimic. You should be able to say "right" just like Jeremy or "turkey fucker" like Johnson from the first go.

I find that I have to learn how the phoneme is actually said (where is the tongue, etc.) and then practice it a lot with recording(s) of native speakers (i.e. a recorded TV show) to compare with. Once you can make the right sound, practice it until you can say it easily.

If you have friendly native speakers, you may be able to get them to exaggerate the sound. E.G. the difference between thin and then is that the 'th' in then gets voiced (your throat vibrates) and the vowel is different. 'World' has an extra syllable (wur - ul - d) vs. 'word' (wur - d). I'm less sure about your issues with the others, save that certain sounds probably don't exist in your native language.

And those you just have to study & practice. I know how hard it is, though, having learned such sounds as 'ng', the 'eu' vowel in 'feu' (French - fire), glottal stops (yes, those exists in the English word bookcase, but...), the Japanese 'R'-group sounds (they're not 'R's or 'L's as we know them).

As others have mentioned, there are many different British accents. I wouldn't recommend a Cockney accent. However, a nice neutral British accent, like a BBC newscaster is a very good accent to have. Most Americans perceive people with such accents as being more intelligent and more attractive than people with neutral American accents. As someone who grew up in the Midwest and intentionally made my accent more neutral American while living on the East Coast, I have a pretty neutral American accent. However, now that I'm living in Hong Kong, I'm intentionally trying to pronounce my vowels in a slightly more British manner. However, I'm too attached to my Rs to switch to a non-rhotic British accent.

A slight Southern American accent sometimes does cause many Americans to perceive someone as more trustworthy, and some Americans do find a slight Southern accent to be very attractive. Though, be warned that a Southern accent may cause some Americans to perceive you as more stubborn and slightly less intelligent.

Britain has many incomprehensible accents, but as long as you stay in the realm of comprehensible ones you'll be fine.

Yes, there are plenty of American comedy skits about unintelligible British.