I am attempting my first Bitcoin transfer from Abra to Bittrex. I am only sending the equivalent of $10 US dollars to test it. The money did not go through as I do not have enough in my account to pay the fee which is $44!! Is this right - it costs $44 USD in fees to send $10 USD?
Yes, they are that high at the moment. You should not send such small amounts of bitcoins nowadays. You could wait until the Lightning Network works. The fees will probably fall then.
Yes. The price of transactions fees in Bitcoin are not a percentage of the amount you are sending, they are formed from a market process and influenced by their size in bytes.
As there is a limited amount of space in each block, all users are currently bidding against each other for space in the blocks. As you can see here, the price of transaction fees has been bid up to approximately what you described.