Electrical – Is it safe to install a wire nut inside a base board heater

I really question the wisdom of switching just the lower heating element. Doing so is going to overwork the upper element. I assume you are trying to conserve electricity, but this tactic is not going to be a great benefit to you. The upper element is used as the "quick recovery" unit that gives the cooling water from the lower part of the tank a boost as cold water is added as the hot water is used. If the water cools too much in the tank and you call for hot water, the upper element will not be able to keep up until the temp of the majority of the water warms up. I would suggest an inline switch or timer controlling both elements to help conserve energy and insulate the tank as much as you can. If you are looking for short, small quantities of hot water , you may consider a 1 or 2 gal point of use electric tank, in line at the specific appliance. There are several models to choose from at the box stores for under $300.00 I might add, if you do try this switch idea, you will need to switch both 120VAC lines with a DPST,(double pole, single throw) 30amp switch wired with 10AWG copper. When you open the thermostat cover, you will find the lines are connected to screw terminals on the thermostat switches.

I would avoid using the baseboard ground for the outlets since the baseboard is on a different breaker (circuit). Per NEC:

NEC 300.3(B) All conductors of the same circuit, including grounding and bonding conductors shall be contained in the same raceway, cable, or trench.

You can certianly replace the home run and associated branches with 3/12 (20 A) or 3/14 (15 A) depending on the circuit breaker size.