Sadly, Jungles are counted different from forests in IV. They're actually worse than useless.. They do not give you any lumber, cause unhealthiness (malaria?), and require the very expensive Iron working to cut down. For the most part they're usually very harmful for you cities; while IMO they should have acted like floodplains (more food but in exchange for unhealthiness.) The only good thing about jungle is that they usually hold calendar resources and thus is good plantation/farmland after you chop it down.
It's for this reason why any start near jungle blows so hard-- you can't even use most of those tiles til later on.
Overall, happiness is definitely the more limiting of the population caps, and thus we usually don't bother too much with health. The reason for that is that unhappy citizens don't do anything but waste food and cost maintenance. Usually you'll have enough food surplus to overcome unhealthiness until later on. If the city doesn't have that much food to begin with, unhealthiness won't be a problem. And if your cities are truly unhealthy and unhappy, just whip them down.
And a final note to why you should chop down forests is that they are liabilities during war. If the enemy sets up on defensive terrain on your land, you'll have a lot of trouble. There are some gimmicks like using forrested forts at narrow spots but in the majority of wars you just don't want your enemy securing themselves in a nice spot.
An exception is probably the national park.
Well, you could save trees in tundra areas since tundra tiles are so bad anyways.