Many common rocks have at least some attraction to a magnet. Native iron is a rare terrestrial material and only found in particular locations, but anything having significant amounts of the iron oxide mineral magnetite can be strongly magnetic. Usually such rocks are principally hematite with secondary magnetite and will leave a characteristic red/brown streak on a porcelain tile though.
Weaker magnetic attraction is a common property of a wider variety of rocks (many of which will leave no streak), notably those in the basalt family, from minor amounts of magnetite plus minerals such as ilmenite or ulvöspinel (both iron-titanium oxides). I don’t think it can reliably said what your rocks are without close examination in hand, but (for example) these are basaltic pebbles (picture is copyrighted, so I am only providing a link):
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If you Google the term you’ll see lots of other examples. Weird marks like those on your first rock can arise in all kinds of ways… from shrinkage during cooling of igneous rocks, from invasion or differential crystallisation of secondary minerals as web-like veins which then erode from the host rock during water transport and from surface scarring during glacial or glacial meltwater transport of cobble and pebble-sized rocks. Those kinds of marks are absolutely not characteristic of meteorites though.
It’s true that most meteorites are attracted to a magnet but because that only applies to “most” it’s not a conclusive test. Some chondrite meteorites in the ‘LL’ class (Low total metal/Low native metal) are only barely attracted to a strong magnet, carbonaceous chondrites usually not attracted, and stony achondrites rarely attracted (but never strongly so). However, what can be said with 100% certainty is that if a rock has a metallic appearance or exhibits any veins or flecks of metal and is not attracted to a magnet then it isn’t a meteorite.
A meteorite doesn’t have to be cut to confirm it as such, and certainly doesn’t need to be cut in half. There are many non-destructive tests (or at least which leave minimal noticeable marks) that an expert laboratory can perform for confirmation. Cutting is however usually necessary for typology classification. Even then, a small ‘window’ is usually all that is needed, not a full cut through the specimen.