Repairing a notched floor joist
Good on you for leveling up your knowledge and learning where not to drill or saw on joists. On that same note, here's an excellent summary from BuildingAdvisor titled Guide to Notching and Boring Joists:


Don’t make any holes with a diameter greater than 1/3 the depth of a joist.
No holes closer than 2 inches to the top or bottom edge.
No holes closer than 2 inches to any other hole or notch.
No notches in the middle  1/3 of the joist, but holes are permitted here.
No notches deeper than 1/6 the joist depth.
No end notches (where the joist is supported) greater than 1/4 the joist depth.
The length of a notch should not exceed 1/3 the joist depth.
Do not make square or rectangular cutouts. Also avoid square cuts in notches — angled cuts are better, as shown. Square cuts tend to start cracks.
No notches are allowed in the top of a large beam (greater than 4 in. thick), except at the ends.



  I've heard that I can glue/screw a 2x4 to the bottom of this joist to make it an upside down T shape. Will this restore enough tensile strength?


A glued and screwed 2x4 fastened to the bottom of the joist would increase the tensile strength. Enough? You shouldn't trust anyones advice on that but a structural engineer, and I'm a different type of engineer. That's not among the fixes I've seen employed in the building trades.


  Does this need to be fully-sistered (with the pipe relocated)?


That's one option. There are others.


Sister the first unaltered joist, making it a trimmer. Then, temporarily support the 2nd and 3rd improperly altered joists, cut them off on either side of your steel pipe, and run new double 2x8 joists (aka: headers) from the rim joist to your sistered trimmer. Add in the requisite joist hangers, remove your temporary supports, and you're good.
Add flitch plates to the cut joists.
Add a support beam beneath the two cut joists, near the cut. Voila, now the holes/cuts are no longer in the middle third of the span and the span is roughly half.



  How urgent is this repair?



Unless you're planning to park a fridge in that closet, this is unlikely to be urgent.
It looks like your 2x8 is wider than 1-1/2"
A modern 2x6 would suffice for your span, which is what you made your 2x8 into.
Your subfloor looks like 1", those boards will help distribute loads across joists.
The cracks are normal. No big deal. Search on "Checks and Splits in Lumber" for details.
The closet walls are unlikely to be supporting. Supporting walls are atop framing that carries loads down to footings.
Try a deflection test. Have a helper eyeball the floors deflection as you jump above the cut and uncut floor joists. Does the floor feel more squishy above the cut joists? Does it deflect more?