Given that Czech doesn't have articles there isn't a direct equivalent of 'the' and 'ten' often seems to be used where English would use 'the'. Similarly 'nějaký' is often used where English would use 'a' - for Czech learners of English it often means they overuse 'some'.
the one exception in common, non-standard czech is that where the 1-st person subject pronoun is included, the jsem/jsme may be skipped. so Já ten dopis psal celej den. would be a well-formed sentence (although incorrect in standard czech).
Yes, you need the appropriate "být" auxiliary for both first and second person, in both singular and plural. We got lucky only in the third person. :-)
This letter vs. the letter, is that really that important that you get the sentence wrong? There's no one right translation and that's it. There are possibilities and without context it's harder to guess.
This letter and the letter are two different words (especially this X that) and are not interchangeable.
I think in this case the correct translation is the. This is more frequently translated with tento, tato, toto.
This DL program is quite consistent: 'ten/to/ty' etc is always translated as either "that" or "the", never as "this".
Furthermore, when a bare noun, with no determiner in front of it, appears in the Czech sentence, this program is consistent in allowing you to insert the indefinite article "a" into the English translation, but not the definite article "the". In short, if no determiner in the Czech, then no "the" in the English.
The one exception to the above that I've noticed has to do with possessive constructions. For example, in addition to "the doctor's office", DL will allow (as indeed it should) "the office of the doctor", even when the Czech original has no demonstrative corresponding to the first "the" in "the office of the doctor".
THE and THAT often map into ten [ta, to] On the other hand THIS often adds some visibility touch to the meaning. All translation is contextual, for example THIS in past narration often means A/AN [..and so then appeared this man and ..]
No, the vowel is shortened in the past participle. It happens a lot in verbs with monosyllabic infinitives but not in all of them.
See "Czech - An Essential Grammar" (Naughton), which you should be able to download free. At Ch. 7.5.3: "The long vowels of monosyllabic infinitives usually short in the past -l forms." There is more, but psát is one of the verbs where this occurs.