i a. I like to stay home at weekends. b. I like staying home at weekends.
ii a. I’d like to be a politician. b. I’d like being a politician.
With [i] there are many contexts where [a] and [b] would be equally appropriate, but there are also some favouring one or the other. Suppose you ask me to go bushwalking next week-end but I wish to decline: [a] would here be more appropriate than [b]. Conversely if I am currently enjoying a week-end at home [b] is more appropriate than [a].
The infinitival is more associated with change, the gerund-participial with actuality. Thus someone who has recently turned forty or got married might say I like being forty or I like being married. An infinitival would be strange here, suggesting repeated changes from not being forty or married to being forty or married. In this case the meaning is close to that of enjoy, which only allows gerund-participials. Would like, by contrast, projects into the future and resembles a verb of wanting, with a strong preference for the infinitival, as in [iia]; [iib] is possible, but the interpretation is roughly “I’d like/enjoy the life of a politician”. If we change the examples to I’d like to start the meeting a little earlier this week the gerund-participial becomes quite implausible: I’d like starting the meeting a little earlier suggests that the starting is itself something to be enjoyed, which is an odd idea.
Hate with a to-infinitival has an idiomatic use seen in
 I hate to tell you this, but your battery is flat.
This can be thought of as involving projection into the immediate future: “I’m going to tell you, though I hate having to do so”. What is special about this use (virtually confined to the 1st person) is the combination of simple present tense in the matrix and single dynamic event in the complement - constrast [52ia], where we have repetition of staying home; other verb of liking and not liking do not allow this pattern, though it is found with adjectives: I am happy / ✲like to tell you that you’ve passed your test (cf. also regret in  below).
(The Cambridge Grammar of the English Language)
Question 1) “suggesting repeated changes from not being forty or married to being forty or married” -> what does this mean?
Question 2) “In this case the meaning is close to that of enjoy” -> What ‘case’ does this sentence indicate? Is it an infinitival (I like to be forty, I like to be married) and the infinitival means ‘enjoy’?