The news is very strange to me. May be Chinese wanted to see reactions from Nepal and specially from India by spreading such funny news. Usually chinese do that type of opinion gathering mission.
1. Mount Everset is not above kathmandu between the border of Nepal and Tibet. Mount Everest is in Sagarmatha zone.
2. If they want to dig a tunnel below mount Everest to reach Kathmandu the tunnel will be long, expensive and useless.They can reach Kathmandu fast by using Kodari high way or other highway vai Rasuwa.
3. There is another highway under discussion which reaches Rajbiraj or nearby from Chinese border.

4. On Chinese side there are many roads, they just have to build north south roads in nepal.One is already there.One is under consruction and third one is under consideation.
If the Chinese want to reach Delhi,faster building tunnel under mount Everest is useless because Delhi is closer from western Nepal and Mount Everest is in eastern Nepal.

5. Only benefit from digging a tunnel below Everest is geological investigation of area for different minerals , and fossil fuel/gas etc etc.

6. Watching Indian reaction is even funnier. India China share more than 3000 km border and china has build heavily on chinese side of the border, they dont need Nepal to spy or attck India.
7. China have satellites and other devices spying on India.

8.There are lots of Indians who will sell any secrets if they get good money.

9. This is not 1975 or earlier where one country interestd to grab the land of another country, Now economic stability is important that land

10. Being transit for goods between India and China, nepal can make only be arround 100 to 200 million not more than that. we have to be very carefull here. If the transit is by road, there will be thousands of trucks on road creating environmental problem.

11. Transit using trains is not that bad but nepal will get only samll share of transit route.
shipping cost of one Container from Shanghai to Mumbai by sea is approx 2000$, and transit time is about 23 days. Via Nepal transit time could be only 10 days. If the cost of shipping is same then nepal will get only about 200/container. If 1000 containers/day shipped via nepal .I guess only 10% of land route from shanghai to mumbai lies inside Nepal.
12. 1 train carrying 100 containers, ( 100, 40 ft long containers makes one mile long train if gap between containers also included). New train every 2 hrs, 24/7/365 day soperation with no down time Nepal will get about 73 million.

13. This is rough calculation but plausible
14. If someone is expert please dont hesitate to share