Power outage before Mac finished updating ... What should I do?

My Mac was in the process of installing updates via Apple Software Update. For this update session, a restart was required - i.e. a message box popped up and said I had to restart to finish installing updates. So I clicked the button on the message box that would cause the Mac to restart.

After clicking the button, and before actually restarting, my Mac logged out of my account and went back to the default "space" desktop image where it finished some more updating (i.e. there was a message box with a growing blue bar representing the percentage of the update process that was finished).

Unfortunately, the blue bar had almost reached the end when my Mac suddenly shut down. It had not restarted, but actually shut down, and I couldn't turn it back on again. I checked my phone, microwave, refrigerator, which had all also turned off, and realized there had been a power outage while my Mac was updating.

After I got the power back on and turned on my Mac, the Mac seemed to start up normally and work fine. I tried Software Update again because I thought it might allow me to fully retry the update since it didn't finish fully due to the unfortunately timed power outage. However, it said that my software was fully up-to-date.

I have tried a few programs out so far and it seems to work fine for the most part. The only thing I've noticed is that Safari has been acting a bit strangely. For example, the cursor no longer changes into a hand over hyperlinks; instead it remains as an arrow. Second, whenever I click on any search result on Google, I get the following message (in this example, I clicked on the first search result for my search for "test"): "Redirect Notice The previous page is sending you to http://www.**.com/. If you do not want to visit that page, you can return to the previous page." Third, the full version of Gmail no longer works and I have to use the HTML version. I'm not sure whether these are glitches in the newest version of Safari or if they are a result of the computer not finishing the update process properly.

I haven't tried all software out yet, and I don't remember the full list of software being updated, so issues may potentially exist in other software caused by the update not finishing properly.

One other thing that occurred was that the first time I shut the computer down after turning it on after the power outage, a message popped up on the "space" desktop image after logging out, which said it was updating the system cache or something. But now the computer seems to shut down normally.

This might be a newbie question, but what could I do to address this issue? Thanks for any help.

<Edited by Host>