Let me preface by saying that my experience is with ocean-going tugs, and then mostly with rigging/lashing and route planning rather than the day-to-day details of operation. I also don’t have Reid’s Primer of Towing at hand, so I’m going from imperfect memory and the incomplete working knowledge that I have, so if someone with more experience wants to step in to expand or correct, please feel free.
Depending on how strong the current is and how narrow the estuary is, current may or may not be an issue. It is obviously preferable to go in with the flood tide and out with the ebb tide, but for a wide and well-dredged channel with a deep harbor this may not be a huge issue. On the other hand, getting in and out of, say, Bodega Bay in California is probably heavily dependent upon tides (though I don’t think you’d take anything large enough to require a tug out of there). Part of the issue with tides may not be navigation, but onload/offload operations, particularly if it is a RORO or ramp-access vessel where it needs to be at a certain range of height for loading and unloading. Also, some harbors just may not be accessible at low tide, which obviously necessitates entering with the incoming tide and leaving no later than the early ebb tide.
Barges that are pushed from the rear–with, as you note, a notch to accommodate the bow of the barge–are a little easier to maneuver, but can also be kind of tricky in anything but flat seas found in protected harbors. They only come up to a certain size (I don’t know offhand what the max displacement is) and mainly operate within river and cargo channel systems transporting dry bulk cargo like coal or scrap, though I’ve seen a few that make open-water passages. Towed barges are by far the most common for open ocean transport, and are far more economical and flexible than dedicated cargo vessels, although also slower and somewhat less able to tolerate adverse conditions (up to about WMO Sea State 5 or 6). It is possible to have dual tow arrangements, but rare as it requires a lot of coordination and is inherently more hazardous. Side-lashed barges are less common and typically used where more than one tug may need to be used at a time, or where the barge or vessel has to be nudged up to a dock. Because the tug is pretty tightly secured to the side of the barge, this isn’t suited to open ocean conditions.
The operation of tugs and barges in United States coastal waters, Great Lakes, and navigable waterways is regulated by 33 USC 152, 2071 and 49 CFR l.46(n), and administrated by the United States Coast Guard as with all commercial shipping.
I would recommend reading Reid if you’re interested, as it is a pretty accessible introduction to tug operations, though