The electrostatic baundry conditions were derived for static charges (continuity of the displacement and tangential electric field). Does it hold for the situations of moving charges (current flow). And since current relate to magnetic field, does it means that there are also magnetic fields inside an electric devices?
About second question. Yes, electric current produces a magnetic field. Actually there is a magnetic field inside and around a device or equipment. But the magnetic field strength may be various in each case. For example, there is a magnetic field around overhead transmission line wires. But at the ground level it is negligible (for some voltage classes it is small even around the wires). In most cases before calculation or modelling, it is possible to roughly estimate the level of magnetic field. And decide does it required to accurately compute it, if it is negligible.
Is it even for the case of DC current?
Doesn't the current proportional to the magnetic field?
You have to distinguish between the current and the flow. In a first approximation you can consider the flow carrying both positive and negative charge therefore not producing any current. In the absence of the applied electric field, neglecting charge separation close to surfaces, applying a magnetic field perpendicular to the flow, you have the hall field that create a current perpendicular to the flow. If you apply and electric field perpendicular to the flow, you can deformity the current path.
However, the speed of the flow is much lower than the speed of light therefore the electrostatic approximation is still valid.
In any case the magnetic field produced by the current can be relevant in a gas discharge for extremely high currents. It can be estimated by the ampere law.
Intensity of magnetic field is proportional to the current. Intensity of electric field is proportional to the voltage. If I correctly understand previous question - yes. Magnetic field of direct current is one of the components of electromagnetic field, that is not changing in time. DC magnetic field is created by currents that are not changing in time, that are flowing through conductors that are not moving. In case of DC magnetic field it is possible to consider it independently from electric field.
For a perfect conductor the current densities would be infinite, the current layer on the surface infinitely thin, and there would be no magnetic fields in perfect conductors. In real conductors the currents are finite and fall off exponentially into the conductor with a scale called the skin depth. The magnetic field penetrates the conductor and falls off a the same rate. At DC the skin depth is infinite. There is no such thing as DC, as not many things are switched on without changing for more than 100 years (frequency of about 1/200 per year), and nothing goes on for ever. In many cases DC is a reasonable approximation. DC magnetic fields go right through metals, but do interesting things while the bits are being moved about.
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