****edited version as suggested by user guruofguru64****
edits:
1. Charatchi is unattainable in TF Dream Town
2. Spacytchi is obtained by giving a male teen with the Sewing trait either Good or Bad Care
3. There are 24 raise-able characters in TF Dream Town
Yeah, TF Dream Town was marketed so poorly, that not many users were aware of them. Because of this, a growth chart for TF Dream Town wasn't really cared for. It's taken me about a year to understand— to at least some degree— how the aging system works in TF Dream Town, and I finally have a full list of characters that are in TF Dream Town.
As you inferred, the aging system has a lot to do with playing the minigames associated with the Dream School classes. Attending different classes will give teen characters certain personality traits (performer, beautician, musician, inventor, sewing, chef, and pet stylist). Taking a teen character to any one class, and playing (and beating) the minigame at least three times, will give your teen a certain personality trait. ex: beating the cooking class minigame with Painaputchi (or any teen) three times will give her (them) the chef trait. The traits— and obviously sex— have strong influence on how the teen characters age up, however, I think care mistakes play a role as well.
In TF, characters aged up depending on personality traits, and, more importantly, care mistakes. Care mistakes were measured in Perfect, Excellent, Good, and Poor care. In TF Dream Town, however, it seems that care mistakes are measured in either good or bad care, and I don't know just how many care mistakes draw the line between good and bad, but I can imagine that the line is different for each teen character, as it was in the original TF. So, most characters are selected through, hatching a certain sex, giving the teen a specific trait by taking him/her to dream school classes during their teen phase, and by giving the teen good or bad care. So, for example, hatching a male will give a selection of ten characters that your Tama can age into, then, giving that Tama the inventor trait from the robotics class, will narrow your selection to two characters, from there, the character your Tama will age into will be determined by giving it good care or bad care. Most characters can be achieved by this, but some characters are a little more complex to achieve, namely those that have the Beuatician and Performer traits, and I still have yet to exactly figure out how to achieve them. I'll talk about this more later.
I guess I should talk a little about how I achieve good and bad care. Simply put, I achieve good care by allowing no more than two care mistakes, which means I'm super attentive to my Tama. I achieve bad care, by firstly acquiring a desired trait, and then by neglecting the teen for a long time, usually about sixteen hours, and the hours in which i neglect my Tama varies depending on which teen character I'm raising. For those who do not know, a care mistake is accomplished when a Tama demands help, but is neglected. It may say that it is hungry, that it wants to play, there may be a looming skull above it, or you may forget to turn the lights out when it goes to sleep; when these things happen, the care menu icon in the bottom right will become bold, which signals the player to care for their Tama. Neglecting your Tama during this time will eventually cause the icon to lose its boldness, which counts as one care mistake. The time required for the care icon to pass is estimated to be around fifteen minutes.
I hope I explained my theory well enough. I must also say that none of the information in the chart is taken from any official or credited source, they are simply the results of a year's worth of my personal trials and errors; I stand behind my findings. Here is a rough growth chart:
Robots Class
— Male
Good Care: Mametchi
Bad Care: Righttchi
— Female
Good Care: Himespectchi
Bad Care: Hoshigirltchi
Musician Class
— Male
Good or Bad Care: Tacttchi
— Female
Good or Bad Care: Pianitchi
(Here is an example of a character achieved from good or bad care. In this instance, as well as some other upcoming categories, the level of care does nothing to achieve a certain character. Simply training a teen to acquire this trait will be enough to determine \the character a teen will age into. This is because there is only one character that female and male teen musicians can age into.
Sewing Class
— Male
Good or Bad Care: Spacytchi
— Female
Good or Bad Care: Hanafuwatchi
Chef Class
— Male
Good Care: Kuchipatchi
Bad Care: Karakutchi
— Female
Good Care: Patitchi
Bad Care: Amakutchi
Pet Stylist Class
— Male
Good Care: Monakatchi
Bad Care: Acchitchi
— Female
Good or Bad Care: Chouchotchi (Butterflytchi)
Beautician Class
— Male
Good or Bad Care: Doyatchi
— Female *this category is slightly complex because it features three possible characters where as most other categories feature two at most; I think this implies that specific amounts of care mistakes or the specific teen character that is being trained may have strong influence on the aging process.
I. For Nokobotchi or Painaputchi
Good Care: Coffretchi
Bad Care: Tropicatchi
II. For Neotchi
Good Care: Coffretchi
Somewhere between Good and Bad Care: Memetchi
Bad Care: Tropicatchi
(I listed Neotchi separate from Coffretchi and Tropicatchi, because I've only achieved Memetchi through Neotchi, however, this isn't to say that I know for sure that Nokobotchi nor Painaputchi can age into Memetchi. Furthermore, I have no idea exactly how many care mistakes is needed to achieve Memetchi. I'm just confident that: hardly any care mistakes will yield Coffretchi, a significant amount of care mistakes will yield Tropicatchi, and somewhere in between will yield Memetchi.)
Performer Class
— Male
Good or Bad Care: Knighttchi
— Female *this is by far the most frustrating category because, like female beauticians, there are more than two characters to obtain, however, in the female performer category, there are a total of four possible characters that your teen can age into; like, the beautician trait, I believe that specific amounts of care mistakes and the specific teen character that is being trained may have strong influence.
I.For Nokobotchi and Painaputchi
Good Care: Yumemitchi
Bad Care: Furifuritchi
II.For Neotchi
Good Care: Yumemitchi
Somewhere between Good and Bad Care: Kiraritchi or Julietchi
Bad Care: Furifuritchi
(Like in the Beautician section, I listed Neotchi separate from Nokobotchi and Painaputchi because I've only achieved Kiraritchi and Julietchi through Neotchi, but this isn't to say that it is impossible to yield Kiraritchi or Julietchi from Nokobotchi or Painaputchi, I simply just haven't been able to do it myself. Again, I don't know the specific number of care mistakes that draws the line between Yumemitchi, Kiraritchi, Julietchi, and Furifuritchi, but I can confidently separate them according to good, medium, and bad care.)
I should probably also say that I don't think that the personality trait system from the original TF has any affect on the Dream Town edition. By this I mean that the social trait, active trait, gourmet trait, and the state of having no trait, does nothing for teen Tamas in Dream Town. I tested this while trying to determine the difference between achieving Kiraritchi and Julietchi; that is, I trained two Neotchis, tried to give them the same number of care mistakes (maybe with a difference of one care mistake), gave them the performer trait, and trained one to have the active trait, by completing ten games (not to be confused with the minigames that are the Dream School classes), and trained the other to have the social trait, by sending ten messages, however, they both aged into Kiraritchi. Here is the original TF growth chart that I am referencing, I consider this chart strongly reliable:
https://natashenka.ca/tamagotchi-friends-growth-chart/
This encompasses the full list of the raise-able 24 characters featured in the Tamagotchi Dream Town Digital Friend, not to be confused with the original Tamagotchi Friends. I hope this helped anyone who's still looking for a TF Dream Town growth chart, but can't find one. Of course, if anyone reads this, please suggest edits and corrections.
I've seen charts online for Tamagotchi Friends, but none for its Dream Town variant (which seems to follow a different system because of the Dream School function). For example, I've seen a Tamagotchi Friends chart that described Himespetchi as good (not perfect) care only. But, by taking my perfect care teenager to the "robots class," she evolved into Himespetchi! So clearly there are some differences. 
Can anyone run them by me? Thanks!
