So, first of all: There's no need to create an account. The system, and the users, are perfectly happy with you just posting 'anonymously'.
Do you get Spam from people, who do a regular login?
Absolutely, spammers will sign up. I've no idea what the ration between spam from unregistered vs. registered users is, but both happen. It's hard to say, because we don't get very much spam - which is possibly due to the thing that's annoying you at the moment.
Note: Take my assertion about the quantity of spam with a grain of salt. I've no idea, basically. It's only backed by me not observing any spam when it is undetected, afterwards, I (usually) can't see it.
Why do I have to solve a captcha, after going through a login?
For the aforementioned reason: spammers create accounts as well.
Why do I sometimes get an captcha and sometimes not? Can't the system see, that there has never been a spam from my account[...]?
The system issues captchas based on
- Your reputation
- The quality of your content
- Your ability to solve them (I think)
- A bunch of things I don't know about
The rate at which you're seeing captchas decreases as you gain reputation. At a certain level, 5k iirc, the rate drops drastically for Edits. Though they do still show up even then, if you - for example - make a bunch of edits in a row. Once you hit 10k, they basically stop completely, except for very strange looking situations.
So, to answer your question:
What can I do to avoid captchas?
Most importantly: Gain reputation
Think about how your posts look to the robots, think about:
External links (tinyurls, obscure blogs, your own website, ...)
Phrases like "Thanks" or "me too" in answers
Length of content (esp. regarding very short answers)
There's also an official statement as to the rate at which they appear, made by Jeff Atwood on 'How often do captchas appear?' at MSO:
To trigger CAPTCHA while editing / answering
- The minimum window is 5 seconds
- The maximum window is 40 minutes
There is a maximum of
- one edit every 30 seconds
- one answer every 60 seconds
- one question every 60 seconds
To trigger CAPTCHA while doing performing other actions, there are throttles of various sizes depending on the action -- it's mostly a "slow down" speed bump to inhibit potentially dangerous "I'm going to post obscene things as fast as possible!" situations.
edit: some new reductions in CAPTCHA throttle thresholds, if you have > 10k rep:
- for edits -- reduced by two-thirds
- for post submission times -- min seconds reduced by one half, max minutes increased by x2
edit: assuming you are a logged in user and have >= 200 reputation: after successfully completing one captcha, we now suppress captcha for 5 minutes on your account.
It's reasonable to assume that there are many more rules that the community doesn't know about.
There's lots of speculation about captchas, like "do captchas get harder/easier over time?", I'm going to boldly attribute all of them to Confirmation Bias*. As you rightly point out, this kind of system is basically designed to be wrong very often, so that it can be right when it matters.