Unfortunately in America, the younger generation (like me) are no longer taught how to write in cursive (although there are some schools that still do). However, I find that cursive is very important to learn and I highly suggest you try it as a hobby if you want to dig deep into the culture.

Yes, sometimes, but it's not nearly as formal as Russian cursive. Until the mid-20th century, handwriting rules were quite strict. There was a set way to form and join letters, and we were all taught how to do this at school. British cursive was slightly different from American cursive, but everyone within each country was expected to follow the conventions. In fact, you can see this when you look at old documents - clerks up and down the country all formed and joined their letters in exactly the same way. Until the advent of the digital age, business letters were typewritten, but personal communications were generally handwritten in cursive. That isn't the case any more. Handwriting in Britain, at least, is very personal, and most people write how they want. Apart from a few elderly people, nobody uses the standard cursive any more. Some people can't use cursive at all. Some print, some use their own version of cursive, most use a mixture. For example, many people use printing capital letters but joined-up lower case. Some put curly flourishes on their downstrokes, some don't. Some join certain letters, but not others. It's a very individual thing.

Sometimes

It really depends on the person. In general, younger people do not write in cursive (with the digital revolution and all), however older people do. I am a teenager, but I write in cursive unless I am particularly concerned about legibility.

For me, in the US, I use a combination of cursive and print. My children are not being taught cursive at school. I think my eldest learnt some for a week or two. I think their signatures look funny because they weren't taught the joining rules that I learnt as a child.