I don't think you can fit fenders on that frame. That frame ~might~ fit 28c tires. 99% sure 32s will
not fit. Definitely will fit 25c. You might be able to squeeze those clip-on fenders on with tires 25c or less. But the coverage is slight and the BB will not be shielded.
Personally I commute on a 'converted' or urbanized mountain bike. For many reasons it's a great choice, and you mentioned you might already have one. I never ride trails but it's indestructible here in NYC.
A hard-tailed mountain bike can take a rack on the back, which is really, really useful. Especially if you want to go grocery shopping, you can bungee stuff to it, or use pannier bags.
Also, you can find great already-built Mountain and road bikes that have been cluttering someones garage for years. I saw a nice one today on Craigs for $130. Put a rack on it, slick tires, bar-ends or a trekking bar, toe-clips, blinky-lights and you're all set. I ride about 15 miles or more a day on it.
I've ridden a number of 50-80 mile rides on this 1994 Specialized Rockhopper. I have a pair of Crank Brother Candy pedals for longer rides on it. It's comfortable and reliable.
Theft is always a problem. That's why I don't usually bother with a lock. But when I do I use a U-lock and a 9mm cable for the front rim. Remove the seat and bags.
Not to knock your wind out about building from scratch, but, it's a lot more complicated and problematic than you might think! Find a beater for cheap and take your time building up a fixie. Enjoy and welcome!