I found a phrase "I still remember playing while ...", it was a past tense I suppose but why is there not the "to be" for the -ing form? Why is I still remember playing correct without the "to be"?
"Playing" here is not a present participle, it is a gerund, i.e. a verb form that is used in place of a noun.
The boy is playing. (present participle)
The boy enjoys playtime. ("playtime" is a noun and is the direct object)
The boy enjoys playing. ("playing" is a noun-equivalent and is the direct object of the verb "enjoy")