How do I answer?

When replying to an Inspire Me! in your blog, you have the option of replying in your blog by clicking the button, or in the original senders blog by clicking the button. When choosing the latter option, a new window opens up to the other persons blog so you can read it before posting something.

It sounds like you're using FireFox. The "tiny window" thing happens because when you click the button, it opens up the other persons new blog in a tab in the Inspire Me window, rather than as a full window on it's own. Not particularly useful, and annoying as all get out. There is a remedy though.

Go to Tools->Options->Tabs
Change "New pages should be opened in" from "A new tab" to "A New Window".

Alternately, if you don't want to change that setting you can right click on the Blog button and select "Open Link in new window".

Problem solved.

It's personal preference, but the standard seems to be to respond in the other persons blog. It's more conversational and people can more readily see that there's been a reply to an Inspire.

When this was created, there was only the Reply button. The only way to respond was in your blog. There's actually a horribly complicated way of notifying you in your blog if an Inspire you left has been read by the recipient. It's the "Blogged" section on the left side of your training log, above your Friends list. If you ever need a headache, ask Ron how it works.

Let me try explaining...

A leaves B an inspire. 
A will see B's name on the "Blogged" section when someone else leaves B an inspire on the same day afterward.
The more recent B receives an inspire, the higher in position B's name appears in the "blogged" section. 

Am I qualified for a payed position at BT now?