Well, the vocab included with the kanji is mainly just there to help you learn the kanji. Unfortunately, there’s a fair bit more vocab than that, which you’re going to have to learn… somehow. TextFugu mainly just covers the grammar.

As for your second question, welcome to the world of counter words. Sometimes when you count things in English, you use a word with the number that helps you quantify the noun that you’re counting. For example “six pieces of paper”, “nine pairs of scissors”, “a hundred head of cattle” – in these phrases, “pieces”, “pairs” and “head” are counter words.

Now, you only occasionally use counter words in English, but in Japanese, literally everything has a counter word. つ is the generic counter word, which you only use when there’s no better word for what it is you’re counting (or the better word is simply too obscure) – sometimes it’s also used for simple convenience. It’s also used only for numbers up to ten, for etymological reasons – past ten, you use a slightly different one, 個 (こ). Note also that the つ counter uses the kun’yomi reading for the numbers, but most of the other counters use the on’yomi.

Wikipedia’s got a pretty good article on counter words (including notes about weird variations in pronuncuation): https://en.wikipedia.org/wiki/Japanese_counter_word
Tofugu also has a massive list of counter words: http://www.tofugu.com/japanese/count-in-japanese/

You probably won’t encounter most of these regularly, so let me just list some of the more common ones:
本 (ほん) = long, cylindrical things
枚 (まい) = flat things
回 (かい) = occurrences (of some event)
匹 (ひき) = small animals
頭 (とう) = large animals
羽 (わ) = birds and (oddly) rabbits
台 (だい) = machines and vehicles
日 (にち, but sometimes か) = days

Katanas are counted using 本. Note that nouns in Japanese don’t have a plural form, so 刀 on its own could refer to one katana or many. So nine katanas would be 九本の刀 = kyuu hon no katana.