"Kati always paints firemen."
Translation:Mindig tűzoltókat fest Kati.
Because it is not "always Kati" but, rather, "always (paints) firemen".
If you wanted to say that "it is always Kati", then both the English and the Hungarian sentence would change:
"(It is) always Kati (who) paints firemen." - "Mindig Kati fest tűzoltókat."
In the Hungarian sentence, the focus is placed on Kati, so she moves in front of the verb, kicking the firemen aside.
In that case, it would be
"Mindig Kati fest tűzoltókat."
"Kati" is in focus, it is in front of the verb. Also, "Kati" is after "mindig", making "mindig" refer to "Kati".
So we have "mindig Kati" - always Kati, and we have "Kati fest" - Kati paints, it is Kati who paints. Combining these two gives us
"Mindig Kati fest tűzoltókat." - It is alway Kati who does that.
In the original sentence above, "Mindig tűzoltókat fest Kati", "mindig" refers to "tűzoltókat". So, always firemen. When Kati paints, she always paints firemen. She does not paint anything else. There is place for a little variation:
"Kati mindig tűzoltókat fest" - which means the same thing.
Similarly, "Mindig Kati fest tűzoltókat." can be rearranged with minimal change to "Tűzoltókat mindig Kati fest".
In JimmRepp's version, "Kati mindig fest tűzoltókat", the word "mindig" refers to "fest". Always paints. So, if you go into Kati's gallery, you will always find a fresh painting of a fireman. Other painters may paint firemen once in a while but, if you want to make sure you find a firemen painting, you should go to Kati. Because she ALWAYS has one ready.
No, there's no a in the Hungarian sentence that would warrant the use of "the". The accusative case (-t case) is there to define who does something to whom. In this case, who is doing the painting (Kati, nominative, no suffix) and who is getting painted (firemen, accusative, -t). It doesn't have much to do with whether or not it's definite.
The reverse case, "Firemen always paint Kati", would look something like this: "Mindig tűzoltók festik Katit." Notice how tűzoltók loses the -t suffix and Kati gains it. Also this construction needs to use the definite conjugation (since Kati is a certain person). The definite conjugation would also be used in your translation:
Mindig a tűzoltókat festi Kati. - Kati always paints the firemen.