Is it legal to fly an aircraft with no navigation beacons, and only solid lighting?
This answer assumes that the question is in the context of the FAA's jurisdiction.
The red, green and white lights are position lights. They are required from sunset to sunrise by 14 CFR 91.203.

(a) During the period from sunset to sunrise (or, in Alaska, during the period a prominent unlighted object cannot be seen from a distance of 3 statute miles or the sun is more than 6 degrees below the horizon)—
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(1) Operate an aircraft unless it has lighted position lights;

A solid red or solid white light on the underbelly doesn't correspond to a configuration that I am familiar with and doesn't meet the legal requirements.
I have, however, seen some older varieties of rotating beacons that if broken may become a solid red light mounted on the underbelly. Some older models actually have a motor that actuates rotation. If one had a failed motor it could appear as a solid red light.

The strobes are anticollision lights. The answer is also in 14 CFR 91.203 . If anticollision lights are installed, they must be used unless the pilot turns them off in the interest of safety. The most common reason for this is if the aircraft is flying in the clouds and the diffuse strobing lights are a distraction or cause of disorientation such as flicker vertigo.

(a) During the period from sunset to sunrise (or, in Alaska, during the period a prominent unlighted object cannot be seen from a distance of 3 statute miles or the sun is more than 6 degrees below the horizon)—
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(b) Operate an aircraft that is equipped with an anticollision light system, unless it has lighted anticollision lights. However, the anticollision lights need not be lighted when the pilot-in-command determines that, because of operating conditions, it would be in the interest of safety to turn the lights off.

14 CFR 91.205 is also relevant. Parts of it deal with the FAA requirements for both position and anticollision light systems. It states that aircraft may not be operated without position lights. It also states that aircraft may not be operated without anticollision lights installed with some exceptions.
There are some provisions by which a flight may be conducted legally with broken lights so that it may be repaired. One such provision is also found in 14 CFR 91.205.

(11) For small civil airplanes certificated after March 11, 1996, in accordance with part 23 of this chapter, an approved aviation red or aviation white anticollision light system. In the event of failure of any light of the anticollision light system, operation of the aircraft may continue to a location where repairs or replacement can be made.

If you're really interested, and I mean really really interested, the FAA released AC 20-74 which contains a lot of technical data on exactly how these lights are to be built in terms of intensity, color and position. Delving into this document should be reserved for the truly bored and those who manufacture aircraft lighting.