Overview: Modern Japanese
There is some brief discussion of these in the English Wikipedia, in the "taru adjectives" section of the "Adjectival noun" article here, and a bit more detail in the "taru adjectives" section of the "Japanese equivalents of adjectives" article here.
Long story short, the -taru adjectives in modern Japanese are basically fossils, leftovers that have eroded grammatically. These can only be used in two ways:
- to modify a noun or noun phrase, using the ~たる ending: [堂々]{どうどう}[た]{●}[る]{●}[物]{もの} ("a grand thing")
- to modify a verb or verb phrase, using the ~と ending: [堂々]{どうどう}[と]{●}[行]{い}く ("to go regally")
You can't use them to end a sentence, and there isn't any ~て form. In Japanese, these are often called タルト[形容]{けいよう}[動詞]{どうし}, because they only have the conjugational endings ~たる and ~と.
A workaround
A common modern construction that would allow you to string together ~たる adjectives with others, and not fall too far afoul of the grammarians in your audience, would be to use ~として. ~とする is a common way to turn something into an adjectival or adverbial phrase. It's a bit like the English construction, "it goes ~", or "it's like ~". [女王]{じょおう}は[堂々]{どうどう}[と]{●}[し]{●}[て]{●}[礼儀]{れいぎ}[正]{ただ}しい[人物]{じんぶつ}です。 ("The queen is a majestic and courteous person.")
Irregularities: a language in flux
That said, poking around online does reveal that some writers in Japanese are reinterpreting the grammar: for [堂々]{どうどう} at least, I do see cases of [堂々]{どうどう}で and [堂々]{どうどう}だ, which are technically incorrect according to strict style guides. This word, and possibly this whole ~たる class of words, appears to be in flux with regard to how modern writers are handling the grammar. I would recommend avoiding the ~で or ~だ constructions for the time being.
Overview: Classical Japanese
If you're writing Classical Japanese, things like ~たりて are valid. The MS IME does include some features to support Classical Japanese, since this is still used by modern writers -- particularly for high-register, formal writing.
The ~たる ending is a contraction of adverbial と + the copula ("to be" verb) ある. Technically, the terminal (sentence-ending) and dictionary form in Classical Japanese was あり, and these adjectives could end a sentence with the ~たり ending. So the ~たる ending conjugates just like the Classical copula あり.
Here's the conjugation chart from the relevant Japanese Wikipedia section:
未然形 たら (mizenkei, or "negative stem" used with ~ない)
連用形 たり、と (ren'yōkei, or "-masu stem" or "adverbial")
終止形 たり (shūshikei, or "dictionary form" used to end clauses)
連体形 たる (rentaikei, or "attributive form" used to modify nouns)
已然形 たれ (izenkei, or "hypothetical stem" used with ~ば)
命令形 たれ (meireikei, or "command form")
The ~て form used to join adjectives is from the [連用形]{れんようけい} or adverbial form, so たり + て → たりて. Or analyzed a different way, we use the ~と adverbial form of the adjective and add on the ~て form of あり: と + ありて → たりて.