Should I convince villagers to stay, or send them packing?
Solution 1:
There's no "punishment" for letting villagers go, the main gameplay reasons to keep the village "fresh", new people, new experiences, new furniture (the items a villager gives you is often related/from their interior decorating).
There is however the issue of character portraits. If you are nice to a villager for a long time they will eventually give you a portrait of themself which can be used as furniture. Each is unique to each villager and counts as a lucky item, good for large amounts of HHA points and (I believe) some feng shui.
The only way to get all villagers' portraits (an extremely daunting task likely requiring multiple years of daily play) or, more realistically, to get portraits for the villagers you like, is to keep villagers around until you get their portraits and send enough villagers packing to rotate in new villagers who's portraits (or company) you want.
Also note that the personalities of villagers (snooty, sport, smug) determine which public works projects they can request, so to unlock certain public works projects you may HAVE to let some personalities leave so that others of the required type move in.
Generally speaking, let villagers go if you only have a passing attachment to them (or none) and only keep a couple villagers "forever". Keeping too many villagers forever will keep gameplay a bit stale; you won't see all the personalities, species, individual characters or the interiors of their homes. Since it's somewhat rare (one a week in my experience) for them to ask if they should leave, I would recommend taking them up on it unless you really want them to stay, to allow for maximum variety.
Note that many villagers moving in/out will possibly replace trees or flowers (I don't believe their homes ever remove public works projects or rocks). If you have a Police Station this is less of a deal as the items removed will become "lost" in the lost and found at the police station.
Solution 2:
This depends entirely on whether or not you want the villager to stay around. Perhaps they're really friendly and you get along well, so you want them to stick around, or maybe their house is in the way and you don't like them anyway, so you'd rather see them go.
It's not necessarily a bad thing, unless you don't want them to leave, but telling them you don't want them to leave will pretty much always end up with them staying. This is pretty much like the real world, as you said. Villagers will come and go, if you let them, and if someone leaves, within a couple days, you will have someone moving in to replace them. Sometimes, they might even end up moving to the towns of friends you have visited/have visited you, and you can run into them again that way.
In short, let them go if you want them to leave, as you'll get new ones, and do your best to convince them to stay if you want them to stay.
Solution 3:
It depends if you want them to leave or not. I have about 10 villagers in my town, but I only REALLY like two of them. So, those two, I might keep for a very long time. But any villagers you don't really care for or absolutely HATE, you should tell them to move. You want to keep it fresh, you don't want the same people forever.
Occasionally, when they ask, if you tell them to move, they won't. Sometimes, they will actually decline from moving and say something like "We need to bond more." or something of that sort. They think you just want to get rid of them, so they will think you should get to know each other better and think about your decision. This has only happened to me once, but it is possible.
Every villager is different: some are sporty; some are friendly; some are tough; etc. You want to mix it up, so, whenever you can, try to tell them to move. If they do insist upon staying, next time they ask, tell them to move again. They should move away. You want a variety of villagers, not old, original villagers.
Hope this helped! :D