At first it seems as if you accomplish the same thing so you shouldn't build the wealth buildings and run merchants, but build wealth instead. But look at it this way.
Suppose you have a good food resource + 2 farms + 2 mines, then I could support maybe 3 merchants and still slowly build something in the city like a worker or start on some new military.
Working wealth would accomplish the same thing in the short term with regards to monetary income, but once I start using the mines again, I do have the buildings left to gather gold for at least countering that citys upkeep for the remainder of the game. I never manage to run 100% science anyway since that indicates that I have not expanded far enough or it indicates that I have no military.
So in the long run, wouldn't it work out better that way?
Those buildings don't magically "gather gold". They multiply the city's existing

output (other than from Build Wealth). So if you are running no merchants/priests while you build your army after your wealth-building phase, and have only a handful of commerce from your farmed river tiles running at a high science % on the slider, you aren't getting any significant value for your +25%

market.
Unless you can forecast significant

- or specialist-based

production in the medium- or longer-term, or will need the happiness/health, or need banks for building Wall St, the hammers that went into a market/grocer/bank in such a low-

city would have been better spent converted straight to Wealth. Then you can finish your wealth-building phase faster.
@1. I've seen many people say that it is better to build wealth if your science is below 100% and then adjust your slider to more sceince, but when would you build science then? Seems rather odd.
The point is that building wealth or science happens at the same rate, but normally in the early game (and in most games) the conversion of

to

is at a higher rate than

to
To make a 2-city example, you have a "hammer city" that produces H

and will build

or

and the other "commerce city' has produces C

and has only a library.
Suppose the builder city can produce enough

building wealth to just pay the expenses. Now the 100% science slider (call that S=100) and library produce C*(S/100)*1.25=C*1.25 as

.
If instead the builder city builds

, then the slider S must drop to satisfy H=C*(1-S/100) to pay the bills, so S/100=1-H/C. Now the

output is H+C*(S/100)*1.25 which is now H+C*(1-H/C)*1.25=H+(C-H)*1.25=C*1.25 - 0.25*H.
The point of the example is that the same amount of

is produced to pay the bills, but less

is produced in the second case. If that city had a market and no library, the correct decision changes, of course.
Thus there's merit in specializing cities towards

and

. If you're running 100% science, don't bother with

-specialized cities unless you're going to commit it to running lots of merchants and priests (i.e. Caste System).
@3. I see. I often grab close to home resources first and then I get some nice early production cities with 1 food and several mining opportunities. It helps me early game, but some production cities never go beyond 10 population later on since the grounds do not permit it. But by this method I cannot seem to grab a lot of commerce spots. Perhaps I should focus early production on chopping the future commerce cities by putting down cottages over the grassland-forests and use the wood-hammers to build early buildings more?
Yep. A good general strategy is decide that a city will be a cottageville, and plan to chop out most of it into cottages, remembering to have enough

(i.e. a farm per plains cottage and grassland mine) to get all cottages working. Don't whip this city if it stops you working cottages! There will be a quiet time mid-game when this city won't build much of anything except maybe university and bank while its towns grow, then a Levee and/or Universal Suffrage brings it to powerhouse status. This should definitely be the approach in your Bureaucratic capital in most cases.
Grassland rivers are the best city sites in all cases - either for

or

.