@DSMLVNV can you confirm it is a 225a main breaker (hard to read in the pictures). Indeed
@Sophias_dad that is what it looks like based on the model # of the panel. That is really interesting. I don't ever see that around here.
@DSMLVNV can you post pictures of your actual meter? What is interesting is that most meters are either 200a rated, or CL320 (which is basically 400a) rated. I wonder if you really have a 225a service? Or if it really is just a 200a service in a panel capable of handling 225a. Regardless, I suspect you have plenty of available capacity.
You could either do a NEMA 14-50 receptacle (or 6-50 if you did not feel like paying for the neutral wire and wanted to pay the $35 for the Tesla adapter), or a Wall Connector. I personally am a huge Wall Connector fan but it is an individual choice. I ran 3/4in conduit for mine and used 6 awg copper wire and put in a 60a breaker. This lets me charge my M3 using my Wall Connector at 48a (the max charge rate).
Yeah, newer modern code now requires the handles to be tied together on multiwire branch circuits for safety as
@MN-MS100D describes above. This was not required previously. I am not 100% clear though on the details since there are a couple ways to accomplish this. I think the primary requirement is that when you turn one circuit off, it forces you to turn off the adjoining circuit. This is different from when a circuit trips in a 240v circuit. In that case, it is required to "common trip" the other live power leg even if the handle is held in the on position. So 240v breakers have internal mechanisms to force trip the other power leg when either trips. This is different from just adding a handle tie to two adjacent 120v breakers. So I think either method is allowed for multi-wire branch circuits. Most folks just use 240v breakers since they are cheap and readily available.