Can a developing country (say, Nepal) seek advisory and logistics help from the EU to upgrade/modify/modernize its state laws, police force, bureaucratic system, and so on? If yes, what would be EU's response like?
Why don't such countries (seemingly living in miserable conditions) seek such helps?
First I would separate the issue of
upgrade/modify/modernize its state laws from the others. Changing laws is mainly a political problem, not a technical one (there is a part of it that is technical, but you do not need many lawyers to solve that and the best experts are the local lawyers who know the local code system).
Also, it is difficult to transfer the experience with legal codes of other countries, specially if there are significant cultural differences. The fact is that Norwegian laws fit (perhaps not perfectly, but close enough) most of the Norwegian society but they do not fit Nepalese society so well.
So, while it is could be educative to study other country's laws and its effects, that relationship is less one of "developed country has the expertise that the developing country needs" than in your other examples.
A sticky point is that Nepal's ruling class will likely try to retain those laws that ensure its role, so there is a limit to the level of change they will allow in its laws.
Of course, all of the above does not obligatorily means that legal reforms will be forbidden; they are often welcomed but usually their scope is limited. Only in times of crisis reforms are fast (for example, in K Dog's example after the fall of the Communist Block).
Now returning to your core question, technical expertise is given as part of cooperation programs. Just googling "eu technical help development" you get to lists of projects, with details like the ones listed in the very first brochure I clicked on:
Training of leaders and staff on financial management (Mali, 5863).
Implementation of a strategy for communicationinformation- dissemination-to-members (Togo, 5869).
Organisation of on-the-job trainings for young farmers (Madagascar, 5832).
Training of rice producers for improved irrigation techniques and access to markets (Niger, 5672).
Training and coaching of maize and soya producers by local promoters for planning of input supply (Benin, 5677).
Remember that change is SLOW, so do not assume that the absence of dramatic results means that there are no results, or that no help is searched for or provided.
You do not give the kids some PowerPoints about capitalism and suddenly the country becomes Singapore. You may help to educate some farmers, but those farmers must keep working to feed their families so they cannot dedicate full time to change. Infrastructure does not grow overnight, trained professionals need more than a couple of months to become competent, corruption does not dissapear by just announcing a new law, and you do not have the funds needed to change all the country in just "one round"...
Also remember that there are other factors affecting the end result; for example if a country receives help that really improves it but then it becomes affected by a war or a drought then the benefits will be not evident.
The European Development Fund (EDF) is the mechanism that the EU gives aid to developing nations:
Created in 1957 by the Treaty of Rome and launched in 1959, the European Development Fund (EDF) is the EU's main instrument for providing development aid to African, Caribbean and Pacific (ACP) countries and to overseas countries and territories (OCTs).
It is not the only fund, but it accounts for 30% of the EU's development spend.
The money is apportioned by the EU being lobbied to invest in certain areas.
This seems like something the European External Action Service (EEAS) would do. One of their objectives with regard to development cooperation is:
Promote democracy, the rule of law, and the respect of human rights in developing countries
I think countries willing to do as you say can contact EEAS to ask what help they can offer. If they cannot help, I'm sure they can point in the right direction, for example by bringing them into contact with organisations that offer relevant help.
If you have questions for them, I think the easiest is to contact them via their social media.