Learn Icelandic: Ég skrifa stutt bréf og teikna grænt tré. - I write a short letter and draw a green tree.

In this sentence, you are acting on the nouns bréf (letter) and tré (tree), which makes them the direct objects (in the accusative case). Both of these nouns are neuter words. For neuter words in the accusative singular, the strong adjective ending is -t.

What are the base dictionary forms for the adjectives 'short' and 'green'?

Dictionaries list adjectives in their masculine subject form. You will find them as stuttur (short) and grænn (green). We drop those masculine endings and apply the neuter -t ending to get stutt (since the base root already ends in a double 't', we do not add a third) and grænt.

Why is there no Icelandic word for 'a' before the letter and the tree?

Icelandic does not have an indefinite article (the words 'a' or 'an' in English). You just say the adjective and the noun together. So stutt bréf literally means 'short letter', but we naturally translate it into English as 'a short letter'.

Why isn't the word ég repeated before the second verb, teikna?

Just like in English, you do not need to repeat the subject pronoun when connecting two actions done by the same person with og (and). The ég (I) from the start of the sentence still applies to the verb teikna (draw).