Whenever you have some weird electrical problem, and especially after any electrical work, CHECK THE FUSES!
Then check the fuses again.
After that, look for any wires, relays, or ground connections that weren’t put back correctly. Weird stuff happens when ground wires aren’t tightened properly, or something gets pinched under the connector.
The engine starting and not shutting off is definitely in the bad ground category. Disconnecting the battery while the engine is running is a good way to blow diodes in the alternator.
Speaking of the alternator, with everything reconnected, key off. Are the dash and alternator lights off?
With the key on and the engine not running, does the alternator light come on? If not, you’ve damaged the alternator. Time for a real mechanic.
One last thing, CHECK THE FUSES.
And post your results so it helps the next guy.