If the alcohol mentioned causes intoxication, then it is najis. If your hand and/or body come in contact with a najis thing, then you need to purify them in order to perform the wudhu.
Zakat is not obligatory on the monthly salary. Instead, Khums is applicable if certain conditions are met.
Based on the question, your mothers house is not your hometown; your prayers there are shortened.
The wudhu is not valid with the shoes on. Praying with the shoes on is permissible if one can place the tips of the two big toes on the ground while wearing the shoes.
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