First, allow me to welcome you to the fun of Shadowrun GMing. It can be a taxing experience in 5th edition, simply because there are a lot of cases where you'll be called upon to make house rules. I recommend reading the exceedingly long but valuable thread on House Rules,
found here. In any event, I'll do my best to answer your questions.
1) I use a computer or iPad to track initiative and damage for the enemies. Is there anyone that does the same with Shadowrun? Or does anyone have any tips for tracking the baddies damage my players will inflict?
I use a laptop for tracking everything. The way I do it isn't ideal, but fortunately there are lots of different techniques that can be used. Personally, I use an Excel spreadsheet coded to help coordinate initiative on a grid. Color coding tells me who has Initiative left, and each column is a separate Action Phase (also called Initiative Pass in several parts of the book). However, since you're using an iPad,
this might be helpful. It's an initiative tracker for iOS made by a member of the community.
The GM's Toolbox subforum has a LOT of useful tools, actually. You'll find the GM's Toolbox
here. I'd recommend looking through the various dice rollers, the online collection of resources, etc.
2) I know there is no mention of using minis, but has anyone done this or is it all theater of the mind? I ask for purpose of cover and such.
I've done minis before in 4th edition. Sadly, our miniature aficionado no longer plays Shadowrun, or we'd still be using them. He found creative ways to kitbash several parts to make a whole, and he was quite talented at it. I think Shadowrun lends itself well to a board game scenario. In fact, there was talk at one point of building a miniature-based board game called Sprawl Gangers. I'm not sure how development is going on that project, because things have been quiet.