1. Should my company be from the start a 3D service business or I should make products that I can sell due to my business connections?

If you go the en masse route, why not go for mass manufacturing.
A 3D printed product will have to be especially exclusive and require exclusive retail channels to sell well. That is, unless the customer adds the value through customization. One example: I sell small sets of 3D printed party food picks at 20 euros per set, and that is about the minimal value I can sell it at to have some ROI - therefore, this only works if the products are unique and very customized.

There is always a need for 3D printing services, especially for professionals. However we are picky and will often prefer an SLA model twice the cost over other models. $10k will not get you near a budget for this purpose though, what you could go for to start is buying a used Stratasys FDM machine and do ABS mechanical prototypes. Also look into the Ultimaker 2+/3 because it can work with CPE+ filament which is very strong and delivers stunning quality.

2. If your answer is to make products, what products you would make?
Customized items for very specific markets. Say, fashion accessories for HENRY millenials.

3. Would you buy one or more 3D printers for 10.000$?
For the above, a Wanhao i3 farm would not be a bad idea but also look into entry-level SLS machines because 1. these products will be much more desirable and 2. they are fast.

4. What is the 3D printer(s) you would recommend and what are the applications?
Lulzbot and Ultimaker are the best brands imo for the long term, but to start there are many other options.
Lewihe and Witbox are especially good for flexible products such as shoes, but the Lulzbot TAZ 6 can handle that as well.
For jewelry, plastics are just no good.

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