If you're sending that amount of cash (as opposed to, say, a gift worth $300), there are no laws, restrictions, or reporting requirements that will interfere with you getting the money out of the USA - as long as the cash isn't being sent as a part of a criminal activity (i.e. money laundering) or to a nation which the USA has heavy trade restrictions on (i....
Canadian tax law is much simpler than the US. Canada does not have a "gift tax" either for the giver or the receiver, except for some very special cases.
US gift tax is paid by the giver, not the receiver, so would not be payable by a Canadian on a gift given in Canada.
In the US the giver of gifts has the gift-tax obligation if any exists, not the recipient. Even if letting them stay at the house was considered to be a gift it wouldn't be relevant to the IRS if the giver is Canadian.
I'm not familiar with gift-taxation in Canada.
A donation/contribution to a trust is not considered a gift as long as the beneficiary has a future interest in the gift.
If you are referring to a payable on death account than it is the same as your parents leaving you the monies when they die--except it avoids probate. Note that your parents have a large estate exemption so this would only be relevant ...
There is a 10% VAT in Japan, ask your friend to get the shop to arrange a tax rebates for foreign purchase. Then you need to pay the customs duty when import into Bangladesh, it can be arranged by the shipping company.
However, after looking into the Bangladesh import tax table, Violin is under the following code
9202.10.00 String Musical Instruments ...