No problem warming the extract in hot tap water. Big problem if you would use the microwave though. Extract will cook rapidly in the microwave, either the jug will explode or burn.
The half full jug can be filled with hot wort when adding half the extract near the end of the boil. Stir the hot wort into the remaining extract with the handle of a long wooden spoon. Super easy to pour and cool enough to hold safely by the end of the boil.
I'm not recommending everyone do this, but I actually turn the heat up when adding the thinned extract at the end of the boil. The whirlpool I keep going throughout the pour cools the wort rapidly. I can be back at full boil three minutes after adding the extract. The whirlpool collects undissolved extract at the center of the kettle. Be sure to stir in the center also if you do this.