@Etonkovidova This is all yours to grab. I made a ticket with checkboxes because @Mooeypoo said you liked those! :)
Let me know if anything is missing here. I'm sure I missed somethings. The extension should be available on all beta cluster wikis. The messages may not all be translated yet.
Not many templates on beta have TemplateData so you might wanna add it to some templates for your testing. If you need help with TemplateData, I'd be happy to give you show you some examples and point you to docs.

(Need to re-check) (1) The list of suggestions when 'Cite' search term is entered includes reference to a template page that does not exist in betlabs:

(Re-check in testwiki) (2) In production template data (i.e. template description ) is placed on a subpage - e.g. https://en.wikipedia.org/wiki/Template:Cite_book/TemplateData. Since template description comes form Template data, the Template Wizard suggestion list will be fine?
Another example - https://en.wikipedia.org/wiki/Template:Track_listing#TemplateData - the link there will refer to just Wikipedia:TemplateData page.

(the solution will be based on community reaction) 3) When a user enters incorrect data format VE will give a plenty of warning, but with TemplateWizard the template will be just accepted.

Checked for the following requirements:

Default text from Templatedata is shown as placeholder (Example text if no default): T191268 (and comments)
Auto-value text from TemplateData is pre-populated in the field when the user adds it.

There are minor issues that could be specific to TWTest template - @Niharika, please take a look to decide which of them need to be followed-up:
(1) clicking on info icon and when it opens, scrolling down will add some vertical and horizontal scrolling inside the box and eventually a blank box will be displayed:

(2) The last parameter info cannot be viewed:

(3) Windows Edge17 does not display info icons:

(4) TWTest 'age' field does not have validation. Is it just for the test?

@Etonkovidova For #2, which browser do you see this on? I see this:

For #4, yeah it's only on test. That is a numerical input so it can accept negative numbers. Hopefully users don't use it for fields like age in actual templates. I put it in there only for testing purposes.

I've filed tickets T203559: [BUG] Field info popups get weird if user scrolls while popup is open and T203564: [BUG] Windows Edge 17 and IE11 do not display info icons for the other two issues.

In T203052#4559464, @Niharika wrote:

@Etonkovidova For #2, which browser do you see this on? I see this:

It flips upwards if there isn't any space below but it doesn't flip if it's already open though. I see this on Safari.

I re-checked it - and yes, the issue seems to be limited only to TWTest template and it happens only to that template only in Chrome, which makes this bug essentially invalid.

I also found a curios case in betalabs when attempting to insert template CURRENTISOYEAR which is listed in suggestions, makes the template popup non-interactive when each click produces the following error in the Console:

Uncaught TypeError: Cannot read property 'focus' of undefined
  at mediaWikiTemplateWizardTemplateForm.mediaWiki.TemplateWizard.TemplateForm.afterAttached
  at mediaWikiTemplateWizardDialog.mediaWiki.TemplateWizard.Dialog.showTemplate
  at mediaWikiTemplateWizardSearchForm.mediaWiki.TemplateWizard.SearchForm.showTemplate
  at mediaWikiTemplateWizardSearchField.mediaWiki.TemplateWizard.SearchField.onLookupMenuItemChoose 
  at OoUiMenuSelectWidget.oo.EventEmitter.emit 
  at OoUiMenuSelectWidget.OO.ui.SelectWidget.chooseItem
  at OoUiMenuSelectWidget.OO.ui.MenuSelectWidget.chooseItem
  at OoUiMenuSelectWidget.OO.ui.SelectWidget.onMouseUp

I could not find another example of such template - so, if this issue is an issue specific to betalabs setup - it can be ignored.

Nice catch, @Etonkovidova. I can replicate it on beta labs with other templates without any parameters but I cannot replicate it on testwiki. So it seems like the problem is specific to beta labs.

Whoops, it looks like we may have overlooked the focus when there are no fields. I opened this T203722: TemplateWizard: Only fallback to focus on 'add all' button if it actually exists to follow up.

@Etonkovidova Moriel fixed the bug you found above with TemplateWizard becoming unresponsive with blank templates (T203722). Can you still reproduce it?