For question 1, I would rule out option 2 (slider) because many people have problems with their body image. Sliding a slider to the right gives someone a perception that they are fat. People of extreme size (one way or the other) may have a greater sense that the website is not helping them, even more than if a list of sizes on a clothing site does not include theirs. With clothing sites, I am not sure if people deliberately visit the sites of stores that do not include their size, knowing that there are a great deal of clothing stores that would have their size. My point is, I think it's more of an issue for a generic body-measurement app which caters to all than for a clothing store that caters to a specific audience.
While I don't have sources on this question specifically, About Face 3 has a section on Ethical Interaction Design, which says this:
"Products shouldn't harm anyone, or given the complexities of life in
the real world, should, at the very least, minimize harm."
The authors go on to list five types of harm: interpersonal, psychological, physical, environmental, and social/societal. Someone's size not being available in the slider may cause them interpersonal harm ("loss of dignity, insult, humiliation") and/or psychological harm ("confusion, discomfort, frustration, coercion, boredom"). The User Experience Professionals Association's Code of Conduct contains a similar "do no harm" statement.
(As a side note, Peter Morville and Louis Rosenfeld give a full chapter to ethics for information architects in Information Architecture for the World Wide Web.)
So that leaves your other two options.
I tested option 3 (input type: number) by entering a non-numeric value. The value stayed there. I think that would be more user-friendly if you would change the value when input focus is lost, if the control will allow it.
Option 1, I think, is too restrictive as it merely ignores non-numeric input and users can't see what they typed. Ideally, for me, non-numeric input would show up briefly on the screen and then be corrected to an appropriate default numeric value like zero. JohnGB's answer at Is it good practice to only accept numeric keystrokes in number-only fields? speaks to this.
Ideally, I'd pick another option between option 1 and option 3, as JohnGB's answer linked above would indicate. If that's not possible, I would pick option 1 or option 3 and just make sure the error messages (if 3 is chosen) make sense to users.
For question 2, alternatives with good browser support, that may be more of a development-related question. If no one here has an answer for it, Stack Overflow may be a better place to ask. An answer to this question at SO mentions a number polyfill shim for incompatible browsers: https://stackoverflow.com/questions/17976393/html5-input-type-number-not-working-in-firefox. (I haven't tested that solution myself.)
For question 3, I was considering recommending that you choose an "average" value. However, that would allow users to continue through the setup with a value that is probably incorrect. It also raises the body image / ethics concern again to me because of people's perceptions of what is average. 50% of people (minus those who actually are the average measurement - less likely in cm than in some other methods of clothing sizing) who are, by definition, bigger than the average person, run the risk of feeling insulted by that.
So with that in mind, I would recommend for question 3 that you pick an invalid value as a default. Put in zero, or leave it blank and show an error if users try to continue to the next screen. When users see this form for the first time, the fields should be blank. A negative value won't really make sense and will expose people too much to your web app's implementation model (see also Donald Norman, The Design of Everyday Things and Emotional Design). In either case, make it clear that users have to enter something correct, which ideally is accurate to them so that they will derive maximum benefit from the app, in order to continue.
