I agree with Ant-Green - test it and see. BUT that being said, the way the bat file is written it would be running command 1 then when that sends back a return code, it will start command 2.
Now as for your questions, Q1, the SPID from query 1 may still show up if the connection is still active for any reason. Offhand, I don't remember if sqlcmd logs out when complete or if it leaves the session active but in a suspended state for a set period of time.
Q2, yep - you could see that no problem. if the BAT file is run more than once before the first time completes, then you could have 2 SPIDs for each of the SP's
Q3, yep - you could see the same spid twice (or more) with the same query. When the query goes parallel, this can happen.
Now, my answers above are based on using sp_who2, not activity monitor as I am not a fan of activity monitor, but I am pretty sure activity monitor just calls sp_who2 in the back-end (or a very similar SP).
The above is all just my opinion on what you should do.
As with all advice you find on a random internet forum - you shouldn't blindly follow it. Always test on a test server to see if there is negative side effects before making changes to live!
I recommend you NEVER run "random code" you found online on any system you care about UNLESS you understand and can verify the code OR you don't care if the code trashes your system.