The brake pads say F10BS but they've been really hard to find (and unavailable on Tektro site). Does Shimano (or anyone else) offer a resin version of this brake pad as this one is metal-ceramic?
Tektro has on their website a Brake pad finder tool. Entering your brake model number offers several compatible brake pad models. There are options for both resin/semi-metallic and metallic compounds. The above linked page has itself a hyperlink to an option of buying pads direct from Tektro.
Yes, Shimano brake pads can also be used. Your Tektro pad model numbers that can be used with the HD-R290 brakes are E10.11 & A10.11. Shimano brake pad model numbers that can replace these are B01S and M05. Tektro's higher performance pad option is model number P20.11 which can be switched out for Shimano's G01A, G02A, GO3S, & G04Ti. There are a couple other Tektro pad models that can be used with the HD-R290 (see results of the pad finder), however those models aren't listed as having a like Shimano pad.