I'm reconciled to the fact that any Apple products I purchase in the future will probably be of the, um, semi-vintage variety (due to budgetary constraints). With that in mind, what are oldest-era Macs and Macbooks that would you consider usable for everyday office tasks and Web browsing?
From my own experience: I actually nursed a 2004 iBook well into this decade, before it just got so far behind (and so unsupported) that it couldn't do much of anything. I'm a fan of the Macbook Air (pre-butterfly). My 2013 Mac Mini is still performing well.
I think you're not alone in this. I too will be relegated into buying used Apple products in the future as well. I have a 2011 Mac Mini and a 2014 Macbook Air that I both bought during the good times when I was still making good living and owning the best equipment like fast RAID drives, a nice display and a fast professional router that allowed me to do my job well as well as 2 PC laptops (one was given to me by my deceased uncle as part of his will). But after I became obsolete and got pushed out of my last job, I had to be retrained into a new career which now barely pay, I considered, a proper living wage and while I'm ok with the bills, I no longer have the excess income left for those discretionary spending I used to enjoy. In the past decades, I could just walk into the Apple store extremely confident, chat with the geniuses and then walk out with bags of goodies. Nowadays, I just walk in to the store, look at the prices knowingly that I could not afford them all and then just leave.
It is at this moment that I actually started researching on how I could stretch the life of my macs and considering buying used macs to just get me by and at least make me feel I can still buy a Mac as I had never considered buying anything used in my life. Why would I? I was making the big bucks and spent on $100 meals easily!
Anyhow, the research I did on Macs and by speaking with a lot of people in the Macworld locally and visiting several used Mac stores selling used macs taught me a lesson. There is a booming business selling used Macs and used Macs are useful for many things even in 2019! Even those vintage Macs like the PowerMacs even as some people are still buying them I suppose to keep running their older software in 2019, though now the PowerMac G5 is really dirt cheap. Most of these people say they don't need to use their Macs to go online, web browse, online banking and trading as their iPhones/Android phones do just fine and they prefer that! But for those office tasks; accounting, finance, video and audio editing and printing stuff to the laser printer or inkjet then they would use the Mac.
Here's what I found are the oldest Macs that people would consider buying today in 2019. The Macs need to run El-Capitan and have at least 4Gb of Ram. But I think soon El-Capitan will lose support, so the minimal support would be Sierra if you plan to web browse and log on to secured sites that need minimum El-Capitan or Sierra. Personally, I'm not too worried about web browsing as my free Windows 10 laptop will allow me to access government secured sites for income assistance.
I think your 2013 Mac Mini is more than adequate. If you are looking for a laptop and don't mind slightly underpowered CPU, the Macbook 2010 white is a good candidate. Affordable and upgradable to keep up with the times. They are cheap by today's standards and they can run Sierra and High Sierra.