A Moxon vise is a good solution for clamping vertically.

I’m not a big fan of those imbedded metal stops.

I have one stop buit into the bench which is simply a length of 1/4” thick wood that slides up and down in a slot.

For wider boards I use one made of thin material use a thin (1/8”) piece of wood about 3” wide fastened to a the edge of a 3/4” thick board. That is clamped in the vise and the other end is clamped to the bench. This is handy not only for thin boards, but also wide boards. The only disadvantage is if either your top isn’t flat or the board isn’t flat, the board tends to pop up when you first start to plane.

 

Are you in the US? If so, go to Harbor Freight and get a couple of their brand F clamps. They're inexpensive and have only a small nub on the end to hold the screw end on. Grind or file off the nub. This allows you to remove and invert the end with the screw up near the head. You now have a screw adjusting holdfast. Put the tail in your dog hole over ANYTHING you want to clamp down, including a thin strip to be used as a planing stop. A few twists of the handle, and it's set. They hold like a bear, and can always be as an F clamp when desired. This it the simplest and most effective solution that can be used by any of us.

 

1. If I want to hold something in the vice I have, I am pretty much limited to the depth of the vice. Since there are three steel elements which I can not move, I was wondering if some other method for holding pieces vertically could be used.

You can place the wood to the side of the 3 steel elements. You will probably encounter racking but this can be overcome by placing another piece of wood (the same thickness) on the other side of the steel.

 

Here's an example of me doing just that so that I can cut some tenon cheeks on the end of a cane handle using the same vise that you have. In this case, I just used a small square at the other end of the vise to prevent racking. You can buy or make some fancy gizmos for that, but I just use whatever is handy: offcuts, screwdriver handles, etc.

If a workpiece is long enough, you can rest it on the floor a stool, or a block of wood. You can use your vise to hold it there, or you can creatively use some clamps to clamp it to the bench legs, the apron or the top.

Handscrews are also very versatile for holding workpieces, and their square edges make them easy to clamp to other surfaces or in you vises or even in other handscrews.

These two methods can be used for dovetails. A lot of it depends on the scale of your work. Cutting dovetails in 100 mm (4") stock will be much different than 3/8" (8 mm) stock. For me, chopping mortices requires more support under the work while planing requires me to stop the board from moving lengthwise. When sawing I'm trying to minimize vibration while holding the piece at a comfortable position for that. I adjust my clamping techniques based on the piece and the operation that I'm working on.

You'll also notice a 1x chunk of wood just sitting in the vise in the first photo; I'll clamp a piece a little proud (above) the top in the end vise to act as a planing stop.

You can also Google "sticking boards" for a contraption that will make it easier to work longer and narrower pieces. I've made one from some scrap pieces and occasionally modify it to adjust it to a new or different purpose. Even a crude one will work. It is just a jig; I can make another one if I need to. My simplest one was just a board with a rabbet that used a wood screw as an adjustable stop. Size it for the work that you're doing. Here's how my first one started out. The angled stops encourage the workpiece to be pushed towards the fence in use. It adapts well to different widths for me.

Get creative...

 

If you have an apron on your bench, you can use the above mentioned HF clamps to create a twin screw vise knock-off and a board with holes drilled near each end for the clamps.....

 

clamping options:

Anybody working with a metal vise has the two rods and screw as you have.
It has never been a problem for me.
Have a look at various Paul Sellers videos and see how he is working:

https://www.youtube.com/@Paul.Sellers/videos

on this one for instance you can see the board was on the side of the vise

and

 

For dovetailing etc. larger panels, hold one side in the vise, and the other side clamp to the apron. I have a couple of spacers, 3/4" and 13/16", with a dowel across one end so they hang from the vise jaws to help with racking as described above. Size written in sharpie.

On the outfeed/assembly table, an adjustable stop is used for belt sanding and planing (pull stroke). A (8"?) piece of maybe 7/8"x3" with angled slots fastened to the end with large round head screws. The piece is slid up the desired amount to use as a stop, and lowers out of the way when not wanted. The screw heads are recessed so they don't catch on things.

A board clamped to the far end of the bench can be used as a stop- the work pushes against the near end, & the far end with the clamps is out of the way.

You could add filler pieces to the underside of the bench for clamps to land on.

I use handscrews as shown above quite a lot, especially to hold a board vertically for edge planing.

 

Thank you for great and creative solutions.
For now, I will go for simple screws which will act as bench dogs and I managed to find F clamps which I slightly modified so they can go through the holes on the bench (thanks @Bearcontrare).

For planing thin boards, I will go with the solution @rwe2156 mentioned I got some thin plywood which I can clamp on both sides. As I move forward, I will experiment with other methods as well, but for now I guess my problems are solved

 

And here is a modified solution with the thin board and integrated bench dogs:

 

Glad to hear you were able to find the F clamp suggestion helpful. I've found they come in VERY handy for a number of uses. 8^ D