Applying for undergraduate research - include transcript with bad 1st year GPA?
There are two reasons I want to see a transcript when I take on research assistants:


I want to know what relevant classes they have taken, and how they did in those classes. This (hopefully) tells me something about what kind of basic knowledge I can expect them to have in my subject area, which in turn helps me evaluate whether I have a project that is well-suited for them.
I don't want to hire a student that is struggling with their coursework, because a research position is very demanding in terms of time and attention. A student who is currently barely passing classes should be focusing on that, not taking on a new major responsibilities. (My university has a minimum GPA requirement for student research assistants for this reason.)


Definitely include your transcript with your email. Don't explain the reason for the bad grades straight off - why draw attention to the negative? But ideally, you should be able to say something like "I took 'Highly Relevant Course' this spring and it made me really interested in pursuing research in this area", where 'Highly Relevant Course' is something you've done well in, that is directly related to the professor's research. The idea is to show that you have basic knowledge in the relevant area, and you're also not currently struggling to pass your courses.