1。太极拳是一种老人小孩都适合的运动，不管什么年龄的人都可以（using 可以 is better than 会）练功。 2。上海是一个国际大都市，不管你喜欢不喜欢城市(城市 should be omitted )，上海都好玩。(This one is correct grammatically, but we don't express like that. May be you can say like, 上海是一个国际大都市，不管你喜欢不喜欢，那就是很有意思。)

You use the word in correct way, but the logic of the sentence is wrong. 1. 太极拳是一种不管是老人还是小孩都适合的运动。(correct) 不管什么年龄的人都会练功(It is wrong, it means everyone has this skill, but infect not.) 2. 上海是一个国际大都市，不管你喜欢不喜欢（这个）城市，这都是个事实。(correct) No matter you love ShangHai or not, it is an international metropolis. 上海都好玩，it is the truth, but in this sentence, whether an international metropolis or not has no business with whether it is fun or not, if you write like what you wrote, this sentence has two key point, it will confuse me.

Robin It seems to me that the model is: "Yet (可) X is different, regardless of (不管)... , it/he/she/they still (都) behave in this way." There is a sense of being undeterred or unaffected by varying and varied circumstances. There are three elements: 1. 可 - the contrast 2. 不管 - the varied circumstances 3. 都 - the "nevertheless" or "still" element; the unaffected behaviour For example, 爸爸对我很严，常常批评我， 可爷爷却不同，不管我做什么，在他眼里都是好的。

Hi, here is my answer. 1。太极拳是一种老人小孩都适合的运动，不管什么年龄的人都会练功。 BETTER: 太极拳是一种老人小孩都适合的运动，不管什么年龄的人都可以练功。 2。上海是一个国际大都市，不管你喜欢不喜欢城市，上海都好玩。 This sentence is a little bit awkward. If you use “不管A/B/C/D，都E”, you mean that no matter A, B, C or D, you can draw the same conclusion E. The cause "不管你喜欢不喜欢城市" consist of two possible cases, you like cities or you don't. Shanghai is a large city, so it doesn't make sense to draw a conclusion that Shanghai is a good place to visit if you don't like cities. Hope that can help!^^

Your sentences are ok, but we wouldn't say it that way, Please see the corrections below. 1。太极拳是一种老人小孩都适合的运动，不管什么年龄的人都会练（它）。（去掉“功”会流畅些） 2。上海是一个国际大都市，不管你喜欢不喜欢(它)，上海都好玩。（去掉“城市”会通畅些）