Although you can use a 2x objective, you would get better results from using a reversed enlarger lens or a duplicating lens designed for 2x.
The transition point where microscope objectives become better than enlarger lenses is around 3x-4x. By 5x even a fast enlarger lens is severely diffraction limited and this is where the microscope objectives give better results due to their wider effective aperture.
Most microscope objectives are pretty small. Also on many you can unscrew the outer metal sleeve to reveal a smaller internal body. This can be helpful especially if the working distance is short and you are trying to light the subject. A smaller diameter lens is easier to light around.
You are correct that the lens must be designed to perform all chromatic correction itself. Most older lenses from Olympus, Zeiss, Leitz etc split the correction between the objective and the eyepiece. While this is a fine solution for direct viewing, it means that a camera mount must also incorporate an eyepiece with the same corrections.
Fairly early on, Nikon moved from this model to 'Chromatic Free' (CF) objectives where all the correction was done in the objective. These CF objectives are very suitable for photography.
The other thing to watch out for is infinity vs. finite objectives. Older microscopes used a simple tube wit no optic between the objective and the eyepiece. The objective was designed to produce an image at a certain distance, ad the eyepiece picked up that image and magnified it. These objectives are marked with the tube length, 160mm for biological objectives and 210mm for metallurgical ones. The objective sits iside the top of the lens and the actual image is projected 10mm from the top of the tube, so that corresponds to a sensor to objective distance of 150mm or 200mm respectively (obtained with bellows or extension tubes).
With finite objectives, varying the extension changes the magnification but as you move away from the designed extension aberrations increase and can become problematic. This is especially true with higher magnification objectives.
The other, newer type of objective is the 'infinite' objective. Here the objective does not produce an image but a parallel bundle of rays. A secondary 'tube lens' of 200mm focal length gives the rated magnification and forms and image which is captured by the eyepieces (or for photography, the sensor). Prime lenses typically work well here, sometimes zooms do as well depending on how far back the entrance pupil is.
With infinite objectives, varying the focal length of the tube lens changes the magnification. If the objective has a very large and good quality image circle, then the magnification can be 'pushed down quite far by using a shorter focal length tube lens. As an example, the Nikon CFI60 10x/0.25 objective can be pushed down as far as 5x (on a 100mm or 105mm lens) while retaining good quality.
Other things to watch out for - the aperture should be as large as possible, and for microscope objectives this is given as the Numerical Aperture (NA) rather than an f/ number. Objectives do not in general have an iris diaphragh, they are designed to give optimal results used 'wide open'. Due to the magnification, the resolution is limited by diffraction; good fast objectives limit it less.
Watch out for lenses that require a cover slip between the subject and the objective. Especially at higher magnifications (20x and up) and higher NA (0.40 and up) an disparity between the designed and actual cover slip thickness can have a large effect. Some objectives have an adjustment collar to vary the cover slip correction (including in some cases to a thickness of 0, ie no cover slip). Its easier though to just choose objectives designed to be used without a cover slip (at larger magnifications, that means objectives for metallurgy rather than biological ones).
In generl you want objectives designed for brightfield or for mixed brightfield/darkfield microscopy. Avoid exotic types designed for phase contrast, advanced modulation contrast, etc. Those will be more expensive and not give as good results when used for brightfield photography.
Lenses come with different degrees of correction. The best ones are apochromatic and these tend to cost quite a lot. Below that are 'plan achromats' which are designed to have reasonable correction and a reasonably flat field. below that are achromats and then budget or student lenses which may have a poor level of correction, a small image circle and an even smaller circle of high quality. These are not really suitable for photomacrography.
Pulling all that together, objectives are described by five numbers - the rated magnification, the numerical aperture, the tube length and the cover slip correction, and then the working distance 9from the end of the objective to the focus point on the subject). For example 10x/0.2 210/- WD 16 is a finite (210mm tube) 10x lens with a NA of 0.2 and designed for no cover slip and 16mm working distance. While 40x/0.45 160/0.17 WD 1.2 is a finite 40x lens with NA of 0.45, 160mm tube length, designed for a fixed 0.17mm cover slip thickness and with a unworkably short 1.2mm woring distance. Third example, 10x/0.28 ∞/- WD 10.5 is an infinite 10x objective with NA of 0.28, no coverslip and a working distance of 10.5mm. That is actually the spec for the Nikon CFI60 Plan Achromat 10x objective which I would recommend that you use. The level of correction is better than their older CF finite objectives and you can get a useful 5x to 10x magnification range fro the one objective with different tube lenses.
The Nikon CFI60 objectives from 20x and up have very short working distances. Watch out for that. Nikon do make some Long Working Distance (LWD) objectives, but 'long' is relative and the NA is a bit less. The Mitutoyo M Plan Apo series have very good working distances, high quality and a large (30mm or more diameter) image circle but are rather expensive, especially new.
Remember that microscope eyepieces are 24mm in diameter, 'widefield' ones are 30mm diameter. Most non-student/budget objectives will fill a DX sensor with a high quality image (around 30mm diagonal). An FX sensor needs a 44mm diameter image circle and also needs 1.5x the magnification to fill the frame with the same subject.
Nikon M Plan CF -series lenses seem to have been used for this purpose, any other options for good quality results?
I don't have a spare, but I can recommend a source for a secondhand CFI60 10x/0.25 objective. A Nikon AIS 200/4 makes a good tube lens for 10x (or the AIS ED 180/2.8 for 9x). The Nikon 135/2.8 or 3.5 makes a fine intermediate length tube lens and a 105/4 macro similarly is food for 5x. I'm currently experimenting with a 90mm tube lens for 4.5x.
You will also need an adapter to mount the objective on the front of the tube lens. Adapters from Nikon CFI (M25 by 0.75mm) or Mitutoyo M Plan (M26 by 36tpi) to 52mm filter thread are available, and you can use a step-down ring to mount that on a lens with a larger filter thread.
Well, I hope that helps. You also need to consider stable support, lighting, diffusion, manual or automatic movement for focus stacking, and stacking software - but this reply is getting long enough as it is.