Loaded question, a definitive answer is impossible because as @LasseV.Karlsen has stated SqlConnection is not thread safe so behavior is going to be unpredictable. I attempted similar scenarios in the past and failed. Here is what I think will happen with the parameters in your question.
Does SqlConnection processes queries in parallel?
No, it does not know how because it wasn't designed for this task. Though the fact that it's possible to build a process to use it in this manner is tempting.
will those queries be executed sequentially
Yes. The queries will be executed by the SQL engine in the order received. Though your connection object will probably not know which thread to pass results back to and you'll get the dreaded 'object reference error'.
is there a chance that this isolation level may apply to other queries
Yes. If you change the isolation level of the transaction object assigned to your SqlConnection and then one of your threads attempts to use that connection it will have that isolation level by default. The timing of when a secondary thread will do this is out of your control at this point. You can assign a transaction per command (and thereby attain a unique isolation level as desired) but you'll still have the issue of the connection not being thread safe.