Plane

What has to be done upon leaving or entering the EU when carrying one’s own camera equipment or computer?
When transporting valuable personal items, it is recommended to keep a record proving the origin of the goods to avoid possible import duties upon return. Prior to departure, the customs office should issue an identification document in compliance with the procedure for returned goods. This identification document may be issued by any customs office (If the value of the goods does not exceed 3000 euros, the document can be issued directly at the border. Please make sure to allow sufficient time!). For this purpose, the items in question must be presented and should be described in sufficient detail so as to enable problem-free identification upon return: Photos and specifications of the models or serial numbers of electronic devices are helpful.
Proof of purchase can be presented as an alternative to the abovementioned identification document if the receipt adequately describes the item and shows that it was purchased in the home country (or in the EU). However, an identification document is a much more certain option.
The Chamber of Commerce recomments the use of a ATA Carnet when temporarily using professional equipment (e.g. photography equipment) abroad (also see the detailed information on Temporary use).

Visual artists often opt to transport their works in their hand luggage? What is the procedure in such a case?
In the case of works that are taken to a country outside the EU to be sold, it is technically correct to declare them upon import and to pay import duties (import VAT and customs duties, if applicable). 
If the plan is to return these works to their country of origin, documents should be kept that demonstrate that they are one's own works: A list of the works and their values, an explanation of the reason for the temporary export, an invitation letter from the gallery or the organizer, etc. are helpful. The safest way to transport such objects is to follow the procedure for returned goods (also see the detailed information on Temporary use).

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There are two main points to consider when traveling by air with an instrument: booking a ticket and checking in at the airport.
In Europe, there is no general legislation with regard to luggage on flights (apart from security regulations), so airlines are free to determine their own rules. Nevertheless, the EU has issued some recommendations: the carrier should accept musical instruments as carry-ons (details can be found in the General Conditions of Carriage (GCC) of the respective airline) and no airport tax should be charged for the additional seat booked for the instrument. The airline should mark the instrument to indicate how it should be transported in the cargo hold, e.g. if it is to be transported in the heated part of the cargo hold.
See also the list published by the International Federation of Musicians comparing airlines' policies regarding the transport of musical instruments. 

In the United States, the Department of Transportation (DOT) issued a regulation regarding the carriage of musical instruments in 2015 (Section 403 of the FAA Modernization and Reform Act of 2012 (Pub. L. 112-95, 49 U.S.C. §41724) regarding the carriage of musical instruments as carry-on baggage or checked baggage on commercial passenger flights operated by air carriers), according to which small musical instruments may usually be carried on a flight if they can be stowed safely in an overhead compartment in the cabin or under the seat.
Several airlines have joined forces in the "A4A"; they recognise the value of musical instruments for their customers and have expressly included this in their conditions of carriage. Here you can find the comparative list of the conditions of carriage of "A4A" airlines regarding the transport of musical instruments in the U.S.

Some airlines allow guitars to be transported free as hand luggage in soft cases, which are slightly larger than the standard luggage dimensions but still fit in the cabin overhead luggage bins.
Some airlines require that requests to carry a musical instrument in the cabin as hand luggage be made by e-mail at least 24 hours before departure. Whether permission is granted to do so depends on the capacity available on the flight.
This is particularly important on flights for which small aircraft are used and on which hand luggage is transported free of charge but not necessarily in the cabin. Carry-on cases and other hand luggage are often collected upon boarding and stowed in the cargo hold, which may not be air-conditioned. The very cold temperatures during the flight may damage instruments.

Transport of instruments on an additional seat
An additional seat must be booked for all instruments that are larger or heavier than the permitted carry-on allowance. Size and weight restrictions apply and vary from airline to airline. It is essential to contact the airline before booking additional seats.
Due to general safety regulations, additional seats for instruments must always be window seats. When booking online, it is therefore essential to select the seat for the instrument immediately, at an additional charge if necessary, as a suitable seat may no longer be available when checking in at the airport if the flight is fully booked. In this case, the airline may refuse to transport the instrument despite the fact that a ticket has been purchased.
Booking two additional seats for two different instruments belonging to the same musician can be problematic, as an airline's booking system automatically cancels requests of this kind as double bookings. It is therefore advisable to book by phone via the airline instead of online in such cases.

Transport of instruments in the cargo hold
All other instruments must be transported as checked luggage in the cargo hold. It is advisable to use sturdy cases and to check the instruments in as bulky luggage. It gets extremely cold in the cargo hold and temperature-sensitive instruments should be transported in the air-conditioned part of the cargo hold.

In any case, when flying with instruments, you should be sure to arrive and check in at the airport in good time. You should also bring along a printed copy of the airline policy, especially if it refers to travelling with musical instruments. Not everyone at the counter works exclusively for one airline so the staff at the desk may not know every detail of the conditions of carriage.
You should also try to be one of the first people to board the aircraft (e.g. priority check-in) so that there is still enough space in the cabin overhead storage bins to safely store your (small) instrument. If you have any difficulty, you should always remain calm and respectful and communicate with the staff in a cooperative manner to find a solution.