It all seems like much guess-work is overlooked.
Well, it is good to have an inquiring and critical mind, but some of the concepts you ask about are already 400-years old science.
Of course that shouldn't stop anybody from studying the physics behind it, I'm just saying that you essentially ask about so much that I can't do the steps in great detail. Every quantity I mention has a long history of being accurately measured somehow by many people.
First, let's start with the mass of the Earth. Weighing anything works always in some unit, like the kilogram, or pounds. So you define a standard mass and then if you add some more physics to that, you can measure the mass of the Earth in terms of that standard.
One example of physics used here, is Newton's law of gravitation, as explained in this video. I won't go into much detail, because you're also asking about various other things. Generally to get more information other than the video look for 'weighing the Earth' and/or the Cavendish experiment.
After doing this we know the mass of the Earth in terms of Kilograms. Then, we need to know Kepler's 3rd law, as re-derived by Newton (Kepler's constant had an unknown value, Newton was able to show that it's value is $G M_{sun}$).
Then you take the value of $G$ from the Cavendish experiment, and you can determine $M_{sun}$, if you know what the value of the distance Earth-sun is with the aforementioned Kepler 3.
Great, now centuries pass, the cosmic distance ladder is established in the 1920's-30's, and we know how distant the closest stars are through parallactic measurments.
As the above comment mentions, we can directly measure the masses of those closest stars, if they are binaries. Or if they are single stars, we need stellar evolution models. Those models are created with the mass of a star as free variable. Then colour and temperature are matched in the HR-diagram to find the mass by matching of mass-evolution-curves.
Now we arrive in the 1980's/90's where instruments attain such a sensitivity, that we can see the 'binary wobble' for a single star that has giant planets as companions. The first planet around the sun-sype star 51 Pegasi discovered, was a Hot-Jupiter in 1995. The 'wobble method' until today remains the best, most reliable method to determine planetary masses.
Planetary radii can only be determined if a planet transits in front of his star, then we can effectively see its shadow and thus its size relative to the star. Then again through other means we need the radius of the star (most common: evolution models, better but difficult: Asteroseismology). Once we have that we have the absolute size of the planet.
If you got more questions, let's see if we can answer them.
P.S. The term 'Earth-like' is extremely misleading and means that we only have and Earth-like radius and/or mass. We hardly have any other information about exoplanets at the moment.
Of course there are some lucky spectroscopy results finding molecules and the situation is gonna improve soon with JWST.
