Am I right that if the meet-up was planned, I could say both "昨日紀伊國屋で太郎と会った。/昨日太郎と紀伊國屋で会った。" and "昨日紀伊國屋で太郎に会った。/昨日太郎に紀伊國屋で会った。"?

Yes, but my ears keep saying 「昨日太郎に紀伊國屋で会った。」 sounds like you met him by chance.

In that case, I think it would be safer for me to use

昨日紀伊國屋で太郎と会った。 (Meet-up was planned)

昨日紀伊國屋で太郎に会った。 (Met him by chance)

昨日紀伊國屋でばったり太郎に会った。 (To run into a person (not physically). Similiar to met him by chance, albeit more natural.)

Is that okay?

昨日紀伊國屋で太郎と会った。 (Meet-up was planned) Definitely "planned".

昨日紀伊國屋で太郎に会った。 (Met him by chance) This one can mean both.

昨日紀伊國屋でばったり太郎に会った。 (To run into a person (not physically). Similiar to met him by chance, albeit more natural.) Definitely "by chance". If planned, you cannot use 「ばったり」.

Thanks, masaegu.

I think this sort of runs parallel to English:
"I met him at the subway" could be intentional or accidental (although the use of "met" versus something like "bumped into" carries more of an "intentional" meaning).
"I met with him at the subway" is definitely intentional.

Can I say "I met up with him at the subway"?

Hi, I would like to know if this is the right way to greet someone who you just come across?

はじめまして
How do you do

マクスと申します
My name is Max

今日からよろしくお願いします
Please take care of me from today onwards/Nice to meet you

Is it a little bit long-winded?

マクス　＞　マックス

No, it is not long-winded at all.

If Max is your first name, you do not say 「マックスと申します。」. All you can use is your full name or last name in front of the 「と申します」.

Thanks, masaegu.