Is this form of writing 考える correct? It has an extra stroke which turns the bottom portion into 与. I haven't seen it before and haven't been able to find it after searching. If it is correct, what kind of form is it?
考(father) has similar meaning and pronunciation with 老(old men in family) in its original usage in Chinese. It's written like or in 《說文解字》. The ㄎ part is a cane or walking STICK supporting an old man.
與(give) or 与(the core part of 與) have completely different meaning although some part is written similarly. It was written like or . The 与 part is FOOD or soup or drinks inside a bowl, we hold it with both hands politely to give (あげる).
攷 has its own meaning as knock or strike or hack. And was written like , expressing that a hand makes use of a STICK to strike. This is why it has the same pronunciation as 考. However, this writing becomes rarely used in China, most of the time we borrow 考 to express 'strike' or the similar. So as to 考える in Japanese.
Glyphwiki has as an alternative glyph of 考󠄂, so it looks like more than one person write the character that way. I have never seen that, but perhaps it might not be so rare in calligraphy? (Bear in mind that the wiki page provides no source, though.)
Fun fact: 攷 is an alternative form of 考 and it's still used in Japanese words like 論攷.