"I'm always going by bike" sounds odd, though it's hard to say quite why. The progressive seems better suited to habits of a continuous nature rather than recurrent events. Thus "I'm always riding my bike" seems fine, though isn't applicable in this case. In this case I would use "I always go by bike" as appropriate to the repeated but discontinuous nature of the statement. Your final version sounds good to me (though I'd personally drop the "still" as implied by "another"). Though I'd be happy with "quarter of an hour" (en-gb, see my comment), "fifteen minutes" has a nice symmetry with the rest. In speech the "minutes" could of course be dropped as implied.
You mustn't believe all you read, especially about grammar. Why do people sometimes say that "be" can be either an auxiliary or a "lexical" verb? I don't know what "lexical" is supposed to mean here, but that's what they say. But why?
Well, here's my theory. "Auxiliary" in this context means "helping verb", and sometimes there is a following verb that you can imagine that "be" helps, like the present participle of a verb in the progressive aspect. But sometimes there is no following verb for the "be" to help, in which case it can't be an auxiliary and so it must be a "lexical" verb itself (since as we all know, every complete declarative sentence must have a verb).
Does that make sense? Alas, no, because it's not based on any facts of language, but instead, solely on terminology. Maybe we call the "be" that goes with an accompanying verb an "auxiliary verb", but that doesn't prove anything about that "be". Maybe "auxiliary" is an appropriate term, but maybe it isn't.
I could declare that all nouns beginning with the letter "k" are a special part of speech called "kymniad". Would you believe me?
According to the only syntactic tests I know of for what is an auxiliary, the "be" that accompanies another verb works just like the "be" that doesn't. For instance, auxiliaries are inverted in order with a sentence subject in questions, unlike ordinary non-auxiliary verbs:
You are eating. -- Are you eating?
You were seen. -- Were you seen?
Ordinary verbs don't work this way in contemporary English:
You eat halibut. -- *Eat you halibut?
So, we might expect the "lexical" verb "be' to work like the ordinary verb "eat", right? But it doesn't -- not at all.
Are you a halibut? (not *Do you be a halibut?)
Are you in the kitchen? (not *Do you be in the kitchen?)
Are you purple? (not *Do you be purple?)
So where is the language evidence that there is a grammatical difference between "auxiliary be" and "lexical be"? Nowhere.
Best Answer
It’s a British expression, now used quite generally of university study, that used to be especially common at Oxford and Cambridge. I suspect that it reflects the way in which education was traditionally organized at those universities: a series of set-piece lectures, not necessarily compulsory, and a great deal of independent reading that was regularly discussed with a tutor, either individually or in small seminars. However important the lectures were (or in some cases perhaps weren’t), the bulk of a serious student’s time and energy went into reading, digesting what was read, and writing based on that reading, especially in subjects like history, literature, and philosophy. In this setting reading history (say) would be a natural pars pro toto.