"stop to do something" vs. "continue to do something"

There are a lot of different kinds of infinitive. The infinitive to smoke in

• He stopped to smoke.

(as Colin and Fluffy have already pointed out) is a Purpose infinitive, a kind of adverbial clause answering Why?, which one may introduce with in order to, or move to the front of the sentence,

• He stopped in order to smoke.
• In order to smoke, he stopped.

in order to distinguish it from more common types of postverbal infinitive, like the to smoke in

• He began/started to smoke.

which is an infinitive object complement clause, a kind of noun clause that is the direct object of began or start, representing the activity (or when generic, the habitual activity) that the subject began or started.

Begin and start are also alike in that they can take infinitive complements, like the sentence above, as well as gerund complements, like the one below

• He began/started smoking.

which is synonymous with the second sentence.

However, stop, unlike begin and start, does not allow an infinitive complement, though it does allow a gerund. Thus, any infinitive following stop can't be an infinitive complement clause, and the next likely reading is as a purpose infinitive clause.

Both are common uses for stop, and this little curlicue helps distinguish them. English, and every language, like anything alive, is full of baroque details like this.

What the President was stating was that they continued acting heroically.

When he said

...and continue to do so...

he meant that they were still acting heroically. To do so in this case doesn't refer to in order to do so. It refers to acting heroically.

Continue may take to do or doing. Stop (in this sense) may only take doing, as stop to do means something different, as you say.

Sorry, that's just the way English is.