1.You are losing cities because of culture.Culture is the purple note icon.Basically what is happening is that the other civilizations have a city close to your city which is producing more culture then your city.To combat this you need to build your own culture.
Build buildings that give you culture, buildings that do this that are also useful include Library +2 culture,Monument +1 culture and Theatre +3.
The problem may also be that you are building your cities way too close to the other civs, which make this problem even bigger, especially if you happen to put a your non capitol cities close to the other civs capitol city, which produces +6 culture right out of the gate because of the palace.
Later on you can also use the culture slider to increase your culture but this isn't that good of a idea.There are also a few other ways to build culture which I will let you discover by yourself.
So the solution is build your own culture or dont make cities too close to enemy borders.
2. Sick faces apear when unhealthyness is higher then healthy( The more population you have the higher the unhealthiness, for each point of population (the number next to the city's name) you get one unhealthyness and also for each jungle in a cities radius you get some unhealthyness, I think its 0.4 for each jungle.Certain buildings also produce unhealthyness, like forge, the same thing with the coal resource..
You get healthyness by connecting improved (that means you build the needed "building" on it) food like rice,wheat,corn with your cities by using roads, Some buildings increase healthyness like the granary and the supermarket.
By the sound of both your problems its looking like you are neglecting building buildings in your cities, you cant play the game by simple building units because at a certain point its gonna start hurting your cities and your game if you neglect buildings.
Buildings with the red cross symbol give you more healthyness in your cities.
Buildings with the purple note symbol give you culture, which expands your borders.
Buildings with the yellow smiley face give you happiness, which allows you to get more people in your city without people getting angry