Handkerchief as gift...
 

Is it common in Japan to give a handkerchief as a gift? If so, are they exchanged between friends/family/lovers?

Used or new?

Yes, it's probably much more common than you would have imagined. But remember, it isn't one handkerchief that people give as a gift. It's usually a set of 3 or 5, which is readily available in department stores and others. They come in nice boxes and you have them gift-wrapped.

Between friends, hankerchiefs are good. Family and lovers, not really because you would tend to opt for better items. People receive handkerchiefs when colleagues quit, when friends relocate long-distance, etc. Handkerchiefs are good because they are small and light as you would need to buy many sets at a time for the occasions mentioned above.

Handkerchiefs and hand towels are also standard fare when moving. You buy boxes and boxes of them and hand them out to all the neighbors when doing your move-in greeting.

We have a ton thanks to living in a large building and moving in when the building was first built. Pretty much all the new residents gave gifts to each other. There were only 4 neighbors when we moved in and we gave them all towel and soap sets...
When the remaining 30 some residents moved in they gave us either towels, handkerchiefs, or sugar sets.