The te-form can be a shortcut for the "Please..." requests, so that is why it feels like they are used for commands.
te-forms open a wide variety of extra endings that you can't access from the masu form. One of those forms is the "-ing" (present progressive). Some of the others include "after (verbing)" and "(verb) and then (verb)".
It seems complicated, but there are simple tricks that help simplify the patterns. There are 3 different types of verbs so there are 3 distinct patterns.
FYI, If you want more details on how to change the verbs, here's the patterns:
1-For the ichidan verbs (that end in iru or eru in their dictionary form), you just remove the ru for the masu form and replace the ru with a te. For example, tabe-ru (たべる or 食べる, to eat) becomes tabe (たべ or 食べ) in the masu form and tabe-te (たべて or 食べて).
2-For the irregular verb suru (する, to do) becomes shi (し) in the masu form and shite (して) in its te-form.
For the irregular verb kuru (くる or 来る, to come) becomes ki (き or 来) in the masu form and kite (きて or 来て) in its te-form.
3-The godan verbs is the biggest category. Verbs that are godans have a wider range of endings. (Let me know if you want a couple of tricks to change them.) For the masu form, just change the う sound at the end of the dictionary form to an い sound. For example, yomu (よむ or 読む, to read) becomes yomi (よみ or 読み) in the masu form. The te-form of yomu becomes yonde (よんで or 読んで).
To change godans verbs into their te forms, find the ending character and change it to the te ending:
-u, tsu, ru changes to tte (for example, suwaru (すわる or 座る, to sit) becomes suwatte (すわって or 座って))
-bu, mu, nu changes to nde (for example, asobu (あそぶ or 遊ぶ, to goof around) becomes asonde (あそんで or 遊んで))
-ku changes to ite (for example, suwaru (すわる or 座る, to sit) becomes suwatte (すわって or 座って))
-gu changes to ide (for example, isogu (いそぐ or 急ぐ, to hurry) becomes isoide (いそいで or 急いで))
-su changes to shite (for example, hanasu (はなす or 話す, to speak) becomes hanashite (はなして or 話して))