1= yes because the electronic transmission requires such. No charge in the HVB, your vehicle will not start or run.
2. it is the HVB temperature you need to monitor, not the outdoor temperature. The HVB cells are like humans, when it gets much above 90F, they don't care for it and don't work as well. If you start to approach 100f, you are asking for trouble and the car may malfunction. Ask me about it as mine shut down 4x in Michigan during a hot, sunny day when I needed a bathroom break bad and there was no shady area to park in.. Damn HVB temperature rose significantly just sitting in the sun while I needed personal time.
3. Cord doesn't matter
4. If you are that serious about improved HVB capacity, you need to contact third party vendors. Your dealer will not do such.