"Can you stop running?"
Translation:¿Puedes parar de correr?
You're on the right track with your last example. You can use the present participle after any form of estar, no matter who's doing it. (estás corriendo, estamos corriendo, etc.)
The present participle can be used after other verbs too, like seguir and continuar.
One reason for using the infinitive in Duo's sentence is the preposition de after parar. After a preposition, the verb form required is usually the infinitive.
Voy a nadar. (I'm going swimming, or I'm going to swim.)
Sueño con volar. (I dream of flying.)
Pienso en viajar. (I think about traveling.)
¿Puedes empezar a cantar? (Can you start singing, to sing?)
In this example, "running" is a gerund (meaning it stops being a verb, and becomes a noun). You are asking if you can do something ("stop the") to something else (another noun, which is "the running"). When you have a gerund (verb-based noun), you use the infinitive form -er or -ar in Spanish. So notice that "you are shouting" (shouting = active verb being done by "you"), and "end the shouting" (perform a verb = "end", to the gerund noun "shouting") are written the same way, but are grammatically different. Whenever you have an active present verb, such as "you are shouting," you will use "estas" + "-ando/-endo" form in Spanish (estas gritando). Whenever you have the gerund (noun) form, such as "stop the shouting" you will use the infinitive form "-er/-ar" of the word, such as "deja de gritar."
In addition to estar which is the most common verb used with a gerundio in order to make the progressive tense in Spanish, there are several other verbs which can be used. I don't recall ever seeing parar used with the gerundio. The structure for parar + de + infinitive to mean to stop doing something should be learned with that format. If you try some other format that you haven't seen taught you run the risk of being misunderstood. In some cases, what you are trying to say, turns into something totally different. In this case, your sentence would be really weird. It would mean Can you stop BY running? I would say no, you can't. Can you stop by falling down. Yes.
Here is an article about other verbs with the gerundio pasted from Lengalia, a premium Spanish grammar course that I highly recommend. You can find it on the net.
a. in verbal periphrasis.
Acabar Outcome of an action.
Acabó dándose cuenta de su error.
He just realised his mistake. Andar To be in the middle of doing the action expressed in the gerund.
Anda haciendo los deberes. He is doing his homework.
Comenzar | empezar To begin, to make a start with something.
Yo comenzaba escuchando música antes de empezar a trabajar. I began to listen to music before starting to work.
Continuar | seguir Continuation of an action in the moment of speech.
Son las seis y continúa lloviendo. It's six and it's continuing to rain.
Estar Actions which take place in the moment of speech.
Estoy escribiendo un ejemplo. I am writing an example.
Ir Little by little.
Vamos aprendiendo el significado de las perífrasis. We are learning bit by bit the meaning of periphrasis.
Llevar Continuation of an action, from a moment in the past to the present.
Llevo leyendo el periódico dos horas. I have been reading the newspaper for two hours now.
Quedarse To stay.
Anoche nos quedamos viendo la tele en casa. Last night we stayed at home watching the television.
Salir Outcome of an action.
Si cambias euros a dólares, sales ganando. If you change Euros to dollars, you come out winning.
Terminar End of an action.
Cuando no tenemos tiempo, siempre terminamos cocinando pasta. When we don't have time, we always end up eating pasta.
Venir | andar Idea of repetition.
Ando pensando | vengo pensando en ello desde hace varios años. I have been thinking about it on and off for several years.