What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
Is it correct that everything dry is pure even if it has impurity on it?
If something impure becomes dry, it remains impure and is not purified by drying. However, the impurity does not transfer by touching it if the one touching it is also dry. And Allah the Almighty knows best.
I hired a private tutor for my son through an academy that sends teachers [to families] and takes a percentage [of their pay]. At the start of the [new] school year, I arranged directly with that same teacher, without the academy's knowledge. What is the ruling on this?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The ruling on this matter depends on knowing the specific terms of the agreement and contract concluded with the academy. If the agreement stipulates that no arrangement may be made with any of the students except through the academy itself, then it is not permissible for you to breach the terms of this contract, based on the words of our master, the Messenger of Allah ﷺ: "Muslims are bound by their conditions." (Related by al-Bukhārī)
However, if the agreement pertains only to teaching a specific subject, and the agreement concludes once that subject is finished, then there is no objection to arranging with the teacher to teach other subjects after the contract period has ended, outside of the academy.
Whatever the case maybe, the teacher must uphold the terms of his contract with the academy. And Allah, the Exalted, knows best.
What is the ruling on attending relatives` parties held in public halls, or hotels since if we don`t, we will be subjected to their admonition?
If such parties involve mixing between men and women, or forbidden acts such as listening to songs with sinful lyrics, then attending them is forbidden from the view point of Sharia since people`s admonition is nothing compared to Allah`s, The Almighty`s, wrath.