It looks like I have the answer.
2) Not without difficulties, I managed to find a web-site—ukvi-international.faq-help.com—where one can ask questions about UK visas. The site looks very nice: They offer a free email service promising to answer questions within 3 working days, or a paid (1.4 GBP/minute, waiting time excluded) telephone service. They can even access one's visa application file based on the GWF-number.
1) The response to my question posted above was however in line with the whole UK visa-related business—i.e. provide as little information as possible. It read literally:
1) Am I allowed to use the same visa if I want to travel alone or should I apply for a regular visitor visa?
As long as you have a valid visa with no restrictions imposed on your visa you can travel alone to the Uk.
2) If I travel together with my wife, am I allowed to stay in the UK a few days longer than she does or should we leave the country together as well? Should you have a valid visa to the UK, you can stay in the UK, until the visa is valid. We hope that this has answered your query.
Each time I get in touch with the UK border-crossing-related services, I get the feeling that it's the Home Office' policy to discourage people to deal with them—and thus apply for the visas or enter the country—as much as possible. Not sure how it helps to develop the country's tourist business. May be they feel its blooming anyway and they have too long queues in the airports already (which is true).
In any case, it looks like I cannot re-use the same visa alone, since it states: to accompany xxx. I will apply for a regular visa and ask again once I am at the border. Will post the update then.