Thanks to Tester101's comment on my question I learned about ceramic high temperature wire nuts. I ended up using these: http://www.amazon.com/dp/B0071NCSA4, for 18-14 gauge wire (the ones noted in my comment above were too large, they are for 12-10 gauge wire and didn't hold on to the wires properly).

I read somewhere that halogen lights reach temperatures above 300°C, so the plastic high temperature wire nuts mentioned above are not an option as they are only rated up to 150°C. The WAGO nuts I intended to use were also not appropriate as they are only rated up to 105°C.

While I was already in there, I also replaced the lamp holders with these: http://www.amazon.com/dp/B0055D2Y2E because the metal contacts in the existing ones were corroded.

The result looks like shown below and it works great, no more flickering or humming.

I have to say that these ceramic nuts are a bit iffy to install, probably because they don't have the metal coil inside like the regular plastic wire nuts do. In one instance I twisted too much, and the nut just cut off the wires completely. Another nut never got a good grip on the wires and fell off, so I had to toss it and use another one from the same batch. If somebody knows of a better type of high heat nut, perhaps one that does have the metal coil inside, feel free to comment.

According to the National Electrical Code 2017, the service-entrance conductors should be sized based on the loads being served.

National Electrical Code 2017

Chapter 2 Wiring and Protection

Article 230 Services

230.42 Minimum Size and Rating

(A) General. Service-entrance conductors shall have an ampacity of not less than the maximum load to be served. Conductors shall be sized to carry not less than the largest of 230.42(A)(1) or (A)(2). Loads shall be determined in accordance with Part III, IV, or V of Article 220, as applicable. Ampacity shall be determined from 310.15. The maximum allowable current of busways shall be that value for which the busway has been listed or labeled.

(1) Where the service-entrance conductors supply continuous loads or any combination of noncontinuous and continuous loads, the minimum service-entrance conductors size shall have an allowable ampacity not less than the sum of the noncontinuous loads plus 125 percent of the continuous loads.

(2) The minimum service-entrance conductor size shall have an ampacity not less than the minimum load to be served after the application of any adjustment or correction factors.