"There are five chairs."
One to ten (things counted with つ): 1 ひとつ, 2 ふたつ, 3 みっつ, 4 よっつ, 5 いつつ, 6 むっつ, 7 ななつ, 8 やっつ, 9 ここのつ, 10とお. These all use the "Japanese reading," kun'yomi.
One to ten (plain numbers): 1 いち, 2 に, 3 さん, 4 し, 5 ご, 6 ろく, 7 しち, 8 はち, 9 きゅう, 10 じゅう. These use the "Chinese reading," on'yomi, except 4 is also commonly よん (the Jap. reading). These are used with counters that use the "Chinese reading" f.ex. こ(個) and many others.
I have no idea, but I assume the つ is because of the "counting things" stuff, since that syllable accompanied all the other numbers in the other counting exercises. Also, I know number kanjis have at least a couple ways to be read -onyomi and kunyomi- and one is preferred over the other depending on the situation. For instance, I knew about 4 being read both as よん (yon) and also し (shi), but よん is preferred among other thing because し sounds too similar to 死 (shin = death). I guess for 5 you can read 五 both as ご and as いつ, and maybe in this case the last way is preferred?
The final つ is used to count things. And every numeral kanji has at least two ways to be read -onyomi and kunyomi-. For instance, I know 4 can be read as よん but also as し (but よん is preferred among other things because し sounds similar to the kanji for death. Equally, 5 can be read both as ご and いつ, I guess.
The は particle is a simple subject marker and would be expected if the topic had already been brought up in conversation. It is not incorrect, report it if duolingo does not recognize it in a sentence like this.
However, が can also serve as a subject marker - think of it like a topic identifier, giving emphasis. It's often used if introducing a topic to the conversation, but shouldn't be treated as required.
The same goes for the other numbers - 一 is being pronounced ひと, instead of いち, 二 is being read ふた instead of に, for example.
Jisho says that つ is the "counter for the hito-futa-mi counting system" http://jisho.org/word/%E7%AE%87-1
Does that mean there is a different counter for the Un readings of the numbers? If so, are there particular occasions to use one or the other?
Not exactly, first of all, the 一 in front of it is the number one, so it's not actually part of the counter in any way. Second, 個 is the counter for generally small, round objects, and つ is the counter for just about anything, not just large objects. There are hundreds and hundreds of counters, some used more than others, but these are just two of the more important ones, that are most commonly used.