Neutrals shouldn't be connected together with the GFCI. Connect the neutral of the line and load to the appropriate connections on each side of the GFCI. Otherwise, the current will appear to go out via the hot connection and not return via the neutral on the load side of the connection, which is exactly the scenario that the GFCI detects and trips on.
Also, note this important tip from Shirlock, use the push in connectors for the GFCI outlets: When to use holes instead of side terminals to wire an outlet
Blame glitter obsession
This is a matter of philosophy, rather than pure finance, although builders, being the glitter oriented types they are, tend to treat it as if it were the latter.
Why GFCI breakers?
One of the major problems with GFCI receptacles is that a receptacle in a location can control power to a totally unrelated-seeming location. While this isn't a problem for small appliance branch circuits due to the Code keeping unrelated receptacles off of them, garage, outdoor, and basement receptacles suffer from this on regular basis. For instance, an outdoor outlet might be controlled by a GFCI in the garage, or a garage outlet might be controlled by the GFCI for the wet bar in the basement.
The centralization of all circuit protection into the main electrical panel neatly avoids this problem, as now there's only one place to reset a trip, namely the electrical panel. Another edge that GFCI protection at the panel provides, especially for high power appliance circuits such as the kitchen outlet and laundry circuits, is superior fire protection -- glowing connections often induce ground faults, and a GFCI will react to and trip on this secondary failure, shutting off the power (this is why many AFCIs incorporate a ground fault trip calibrated for higher trip levels than a "true" GFCI).
Another problem with receptacle GFCIs is that they aren't applicable in all cases where GFCI protection is desired -- such as when a circuit other than a 15 or 20A, 120V circuit needs GFCI protection. (For instance, protecting a church baptismal font's immersion heater that runs off of 240V, or protecting the 240V outlets for your garage workshop for that matter.)
Why GFCI receptacles?
However, there is an expectation in this day and age that the bathroom GFCI is in the bathroom, and likewise with the kitchen. The use of GFCI breakers for these circuits is often seen as counterintuitive and inconvenient -- many people would look for a tripped GFCI receptacle in their bathroom, but not look to the panel (which is further away, and possibly harder to get to) for a tripped GFCI breaker. Furthermore, the use of GFCI breakers can make a trip condition ambiguous -- did this breaker trip on a ground fault or an overcurrent/short? A related issue is that a GFCI breaker might conflict with an AFCI requirement, such as for the kitchen and laundry circuits; while DFCI (dual-function circuit interrupter) breakers combine the two protection functions, they are only available in single pole, limiting their applicability.
Finally, GFCI breakers can behave counterintuitively when applied to multi-wire branch circuits as leaks on opposite legs can cancel each other out, preventing the breaker from tripping in a circumstance where one would normally expect it to trip. GFCI receptacles can only protect the legs individually and thus force the neutral to be demultiplexed, avoiding this matter altogether.
It's up to you at the end of the day, but my vote is...
While this is up to you as the Code doesn't care where the GFCI protection lives as long as it's there to begin with, I personally prefer the protection-at-the-panel approach, as long as your panel type and available slots allow for it. It's simpler to make modifications to the branch circuits without inadvertently defeating GFCI protection when the GFCI is in the panel, and it also means that there's no risk of sharing or overloading a neutral on a GFCI-protected circuit as the GFCI protection in the panel will catch that right away.
Best Answer
Finding the first
There's really no substitute for opening boxes, unhooking wires and turning the circuit on and taking measurements.
You open up your believed-closest box, and remove the wires you think are downstream (away from the panel). Tape off the loose wires so they can't short against anything, and plug a 3-light tester or nightlight. Then go turn the circuit on, and see what lost power. If all the other receptacles lost power, you're in the right box. Otherwise try another box.
If the receptacle still has power, you've identified the supply wires. Otherwise touch the loose wires with a non-contact voltage tester and see if you can find the lit-up hot, or just power down and try a different pair.
Usually there are only 2 pairs. If there are more than 2, you may need to disassemble a pigtail. This is also a good time to do housekeeping like make sure your wires are on screws (not unreliable backstabs), upgrade to the $3.00 receptacles instead of the 50 cent builder grade cheapies, and upgrade to 20A (T shaped neutral) sockets if the circuit is indeed 20A and you expect to use those.
GFCI+receptacle devices are very confusing because they are doing two things. Let's break it up.
What is a GFCI?
A GFCI is a black-box which inputs hot(s) and neutral on its "line" side ... and then it outputs protected hot and neutral on its "load" side.
That's it. To signify the protected side, I use brown and gray wire in conduit work, but you can do the same by tagging wires with brown and gray electrical tape. It can be a circuit breaker, or a dedicated black box. If that's hard to visualize, look at this.
This is a GFCI device in its purest form: a "deadface" GFCI. You can only hook this up just like the diagram: supply wires to LINE and downline load(s) to LOAD (otherwise what's the point?)
Since you have first location and supply wires identified, do exactly that. Pigtail the loads if necessary, though some GFCIs use screw-and-clamp to allow 2 wires on a screw.
Be prepared to install a box extension if the box is too small to fit the GFCI device and the wires.
How do you get sockets at the deadface location? Well, heh heh, I don't seriously expect you to use a deadface when you could use a GFCI+receptacle... it's just better for illustration. But here's the gotcha: Since the sockets are pre-wired to the LOAD side, it's possible to miswire a GFCI+receptacle, attaching power feed to the LOAD side - in which case the outlets will be energized but the GFCI will not protect.
How to test for GFCI functionality
Use a simple GFCI tester. Most of them come as part of a 3-light tester, in which case get a simple one**.
You just push the button, and it induces a hot-ground fault which should trip a GFCI protective device. If it isn't obvious, put a "GFCI Protected" sticker on the receptacle.
Now, this external tester will not work if the receptacle is not actually grounded. (there is no way to induce a ground fault without a ground). In that case, get a 3/2 prong "cheater" and extend the ground wire/tab with other wires, to reach a place with an actual ground. Plug the tester into that and it should work/trip.
Once it passes, also put a "No Equipment Ground" sticker there.
Now if this is a GFCI device and testing it also trips another one, then most likely it's fed off that other GFCI's LOAD outputs, which makes it redundant. Remove it and put it somewhere that matters.
** Especially avoid the ones that have digital logic to "help make the lights make more sense": you are much better off with a simple one and interpreting the lights yourself than using the silly legends. The words describing the light combinations, I call them the "magic 8-ball". But the lights are useful.