Your use of quotation marks suggests that you are quoting something that someone else has written. If this is so, you need to cite your source. Failure to do so constitutes plagiarism, and if we allow it, the Grammar Exchange can be subject to legal action.
As it turns out, you appear to be misquoting an article by Jeffrey Hutton, which can be found here:
In Mr Hutton's article, the paragraph actually goes like this:
The notion that pets need more than a minimum of care has been slow to catch on, perhaps because until recently the only animals that came into close contact with humans were those on farms.
The boldface type is mine, indicating where your transcription differs from Mr Hutton's.
Having said that, my answers to your questions are as follows:
1. How is ‘more’ and ‘than’ functioning here? Which word is ‘more’ modifying?
Since you have a compound subject, you need a plural verb: "How are 'more' and 'than' functioning here?". I would say that "than" is a preposition here with the object "a minimum of care", and that "more" modifies the entire prepositional phrase "than a minimum of care".
2. I am I correct in saying that ’perhaps’ as an adverb is modifying verb phrase ‘catch on’.
I don't think so. As I parse the sentence, "has been slow" is the main verb, which is modified by both the infinitival phrase "to catch on" and the entire "because ..." clause. In turn, the "because ..." clause is modified by the adverb "perhaps". Or, more precisely, "has been" is the main verb, which is modified by "slow", and "slow" is modified by the infinitival "to catch on". The entire verb phrase "has been slow to catch on" is modified by the "because ..." clause, which is then modified by "perhaps".
3. Can we use ‘are’ in place of ‘were’. Please explain.
If we were to eliminate the phrase "until recently", the verb "come" would work in the sentence instead of "came", the verb that Mr Hutton actually used, and it would be appropriate to use "are" instead of "were". Since the phrase "until recently" is used in both your version and Mr Hutton's, the verbs need to refer to the past, so "come" makes no sense in context, and neither does "are".