What is a well-established, reputable day-trip tour company in Reykjavik? [closed]

I have done what you planned to do, back in September 2014. There are/were at the time quite a few companies and at that time it was easy to book only on the day before or even the same day for a half day tour.

I did not notice any unreputable tour companies. And I looked for online reviews as well as word of mouth in the hostel I stayed.
And I did notice that I did have much more choice when booking there compared to booking online before I went (as I had checked out and decided not to do.)

As weather is rather changeable, you may indeed find it a better option to adjust your plans to the weather. Going to see a waterfall in very heavy rain is not fun.

As Travel Stack Exchange we do not do company recommendations. And I think that now, after COVID 19 the field may have changed.

I'm actually writing this from Keflavik airport after a two-day stopover in Iceland so here is some up-to-date (if unasked) points

1. Entry is still complicated, you need digital registration and the right combo of test, re-test, vaccination and/or quarantine. However if you are vaccinated you do NOT need to test on arrival and quarantine until test results (as most websites still state).
2. Once you are in it's great! No mask, no doc check, local live is mostly back to normal.
3. This being said, information on what's open and what isn't and at what times is still fairly unreliable. It's best to double check.
4. Be prepared for the weather. Even in July it can be quite cold, rainy and very windy.
5. Credit cards work almost everywhere without problems. We did fine without any local currency.
6. The local people are very nice and friendly and almost everyone speaks excellent English.

I am looking to check-in to the hotel and then explore the city on the first day.

Good choice. Reykjavik is very walkable (provided you are dressed for the weather)

I am seeking some recommendations (name, address, website) for reputable tour operators, with physical store fronts near the city center so I can book some day trips during my 5-day stay.

SE explicitly bans specific product recommendation. IMO your best shot to look at TripAdvisor and Airbnb experiences, and see what's available . The reviews tend to be very reflective of the quality of the operator and you can contact them online, by phone or just walk there if you they list a physical address.