I stopped trying to figure out them running around in non-period clothing in period episodes. I gave up after the Fifth Doctor serial "The Visitation", which took place in 1666. It's just ... not even an issue in the old series, so I don't even notice it when the Doctor and Rose/Martha run around in different time periods in jeans and what not.
As for how Martha doesn't go home, it'll probably be a bit like a reverse-Tegan Jovanka situation. Except Martha really doesn't want to leave, but the Doctor keeps getting dragged into different adventures, either by accident or the TARDIS not making it to where he wants it to go. After a while, he'll probably realize he likes to have Martha around and ask her to stay. That's just all speculation on my part, however.
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As for how Martha doesn't go home, it'll probably be a bit like a reverse-Tegan Jovanka situation. Except Martha really doesn't want to leave, but the Doctor keeps getting dragged into different adventures, either by accident or the TARDIS not making it to where he wants it to go. After a while, he'll probably realize he likes to have Martha around and ask her to stay. That's just all speculation on my part, however.