Can a road bike be "converted" to a hybrid/more upright bike? Flat bars and the shifters/brakes should be simple enough, but will the shifters adapt to my Ultegra set?
Yes, there are no issues with doing this beyond economy and ensuring you use the right controls. You must use flatbar road shifters that match the actuation ratios and gear numbers the rest of your drivetrain needs, but Shimano, SRAM, and Campy have made them for all their mechanical systems for a long time now. (For the electronic groups, EPS has no options, with eTap you can do it with Blips, and with di2 you can just mix and match mountain and road.)
The full conversation of what compatibility differences exist between flatbar road and other flatbar controls across all group and speed generations has become fairly convoluted. Importantly, front derailer actuation ratios have always been different between road and mountain for both Shimano and SRAM. Pre-ten-speed Shimano mountain rear derailers and right shifters use the same actuation ratio that Shimano road has always used (except for 8-speed Dura Ace), but Shimano mountain split off into its own compatibilities completely starting with ten speed.
For the brake levers, you have to get ones that match the cable pull your brakes want. For caliper brakes that means no linear pull (V-brake) levers, and if the brake uses Shimano's Super SLR pull ratio then the new lever should too (the simple clue here if the bike was stock is whether the old STI had under-tape shift cable routing).
I have just fitted flat bars to my Kona Jake. It was running with drops and an Ultegra groupset. I much prefer the more upright riding position, especially in town traffic. I've been able to switch to V-brakes which are much more powerful and responsive than the cantilevers I was using. The gears have proved more problematic. The SRAM shifter I fitted to move the front derailleur is very heavy going and needs a lot of thumb to encourage the chain to get onto the large chain ring. As for the rear derailleur, well, it's a mess. I am using a 9 speed Deore changer, that was knocking around in the garage, on a ten speed cassette. Amateurish, I know, but in time I will sort these problems out. For now, the advantages of upright riding position and more braking power far outweigh clunky gears.