Is it bad if you occasionally trip your main breaker? Like if you forget not to use the toaster and oven at the same time while you are charging?
Yes, you should never be tripping breakers in a properly designed and installed system. Breakers are only a backup safety mechanism. Load calculations are the primary mechanism to prevent fires. Breakers are an additional line of defense. Note that breakers sometimes fail to function!
Thanks a lot. What is the maximum power draw of the wall connector. ( I am thinking 48 amp at 240 volts for a long-range, all wheel drive model 3. Is that correct? What size circuit breaker would one need to enable that?)
The Wall Connector can be installed on up to a 100a circuit, though as others have pointed out, the Model 3 LR can only make use of up to a 60a breaker (48a continuous). Model S/X units can make use of up to a 90a breaker (or older S's with dual chargers can make use of the full 100a).
To verify your dryer is gas, can you turn off the dryer breaker and then operate the dryer? If it is gas, it will be plugged into some other 120v circuit to run the motor.
So I am not convinced that you can't do 60a. Depending on the conduit/wire paths available, I might consider running 6 AWG in conduit (assuming it does not get too hot in an attic - otherwise you need bigger wire) and doing a full 60a circuit. Or you could do 4 AWG romex. You need to look in detail at your load calculations, but your house has as minimal load as possible other than the oven (but I think you can probably still do a 60a circuit). I personally love being able to charge my car at the full 48a and NEVER have to worry about charging being slower than the car is capable of.
Breaker position wise: I am dead curious how many empty spots you have above. It looks like at least one. Hopefully you have two spots available, but if not - I wonder if you are allowed to use a tandem for your Solar backfeed breaker? I did when I consolidated for my Tesla charger install, but now I wonder if that is not supposed to be allowed? You can get Tandems from Eaton in up to 50a. Above that you need full size breakers. As others have stated, if your dryer is unused you could disconnect those wires in the panel and cap them off and then put in a breaker that is say 50a and use that. So you have options! (also you could re-use the dryer wires depending on where they go, but they would only support 30a (24a) charging.
I totally disagree with folks advising to charge slower to reduce wear and tear and energy loss. In a proper electrical system that follows code with proper conductor sizes, heat loss is very low. I don't factor this in at all. I see no reason to believe that Tesla's are not designed to run at 100% of their rated charging capacity every day. This is what I do and it has worked flawlessly since July. Conceptually this is true, but it also runs your cooling pumps, etc, for longer so you have to factor in that wear and tear.
For Time of Use pricing, you often are trying to fit a full charge into a six or eight hour window, and so I just like having max charge rate available to me (even though the math says you can do it with a smaller circuit).
On the cable management for the Wall Connector- I mounted mine outside. I did not use the bracket that can go on the back for Top Entry. I have no problems hanging the cable off the unit. I just use big loops. What you are seeing here are the two loops I take off every day to charge the car and then put back on later. The small loop is slack that stays on all the time. Note that optimally you really want to be able to do this one handed - so the 8' unit in the exact right spot is appealing. I end up leaving the cord on the ground a bit since I need two hands to coil it back up.
Even if you don't go 60a, then 50a I think is totally doable. The nice thing about 50a is that you can do Tandem breakers for 50a and you can use 6 AWG Romex as the feed which is very manageable (only good to 55a since NM cable - romex - is only allowed to use the 60c rating). If you run through the crawl space or attic, Romex might be easier than a lot of complicated conduit bends.
When you send pictures, please also take a picture of the inside flap of the panel with any gory technical details available (sufficient resolution so we can read the fine print). Most importantly getting the model # might allow us to find the specifications.
Let us know what you decide to do! I look forward to seeing pictures of your final install!