I usually ask a dear friend with BA in Japanese, but she's very sick now, and can't answer my questions. You can start with any question you feel like answering.
ありがとう ございました。
First, ありがとうございます can't be used for "thanks in advance," and ありがとうございました seems even stranger. I believe you wanted お願いします. Second, 十一つ and all the others above ten don't exist. It's just 十一, 十二, 十三, 十四, etc. They're read じゅういち, じゅうに, じゅうさん, じゅうよん.
一つ / どの漢字は[なっさる]の漢字ですか。 (なっさる = to do)
It's なさる, and the kanji is 為, but you'll most likely not see it written that way unless you're reading older texts (like from around WWII and earlier). It's written 為さる.
二つ / What's the difference in connotation between 言うand 仰る(おっしゃる) ?
おっしゃる is the 尊敬語 for 言う.
三つ / Is there a volitional form other than たい ?
たい isn't what I would call a volitional form. I'm not sure what to call it, but it means "want to..." What's usually referred to as the volitional form is (よ)う, as in 食べよう, 行こう, and 帰ろう.
四つ / I've seen the word 遊ばない used in the sense "being busy", as in 土曜日も日曜日も 遊ばない. Is this a common use of the form with 遊ぶ ?
遊ぶ can mean "hang out" or "visit," as in "hang out with some friends" or "visit my parents." In that sense, if you aren't going to be hanging out on either Saturday or Sunday you're probably busy. I get the feeling it should be 遊べない, though ("can't hang out").
五つ / What's the difference in connotation between 有るand 在る, both being read あるand having apparently the same meaning ?
The kanji are almost never used, so you don't need to really worry about it, but I would say that 有る has a sense of possession, whereas 在る is more like existence.
六つ / Can you give me an example of the use of たっぷり ?
心配しない、心配しない。時間がたっぷりあるからね。
七つ / Is ですa contraction of で は ありあます ?
であります.
八つ/ I've seen this sentence in a grammar book, お土産の店沢山(が)ありあます, used without が. Is that correct ? How so ?
Seems like it should be お土産の店が沢山あります. 沢山 is used adverbially, so it doesn't get a particle.
九つ / Can どちらbe used for more than two objects ?
No.
十 / What's the difference between どちらand どちらか ? Examples ?
どちらを選びますか。"Which one (out of these two) will you choose?"
どちらか選びなさい。"Choose one (of these two)."
どちら is "which one" (or "which way" or "who" depending on context) and どちらか is "one of (the two)."
十一つ/ An exercise in the the same book said "....に言葉を入れなさい。". What's this 入れなさい thing ? I mean what form is it ? And is the kanji 入 read い or はい?
入れなさい is a command form of 入れる. It means "put in" or "insert," and is read いれなさい. You can tell to read it いれなさい because it's transitive, whereas 入る is intrasitive, so it would be 言葉が入る as opposed to 言葉を入れる.
十二つ/ Another sentence from the book 少々 お 待ち下さい . What's this お for ? And is 待ち下さい more polite than 待って下さい ?
The お is part of the circumfix お...になる that is used to make 尊敬語 out of verbs. お待ちください is more polite than 待ってください, and there is no 待ちください.
Also, there is the circumfix お...する, which is used to make 謙譲語 out of verbs. So お待ちします is a very humble and polite way to say you will wait.
十三つ / Do 遠く ありませんand 遠くないです have the same connotation ?
遠くありません is slightly more formal and polite than 遠くないです.
十四つ/ How do you say twentieth ? 二十? Is it read にとお ?
It's read にじゅう.
十五つ / What are the most common counters ? And how are they usually introduced in a sentence ?
つ is probably the most common, but it depends on what you're counting.
猫が三匹部屋に入った。"Three cats entered the room."
車を三台買うつもりです。"I plan on buying three cars."