près or proches are both acceptable in this case.
Note that “près” is an adverb and “proches” an adjective.

When speaking about physical distance as in here, they are basically equivalent, when talking about other measures of distance, usage favors one over the other, for example “proche” is used for emotional distance, if that makes sense.

Also, in some usages the adverb or adjective is mandatory. e.g. “la fin est proche” (the end is nigh) only works with the adjective.

For the first part, both près and proches can be equaly used as said by @AndroidNewbie.

For the second part vous ne serez pas obligé de marcher beaucoup is indeed correct but IMHO would not be used because it sounds lourd (heavy). I’d rather use:

• vous n’aurez pas à marcher beaucoup (simpler and also correct)
• vous ne devrez pas beaucoup marcher

But it is true that the latter could mean either you won’t have to walk much or you won’t be allowed to walk much and only the context help to understand.

While both would be equally understood, I would definitely use the adjective proche here which “sounds better” to my ear than an adverb.

Tout les endroits que nous allons visiter aujourd’hui sont proches les uns des autres

Endroits près les uns des autres seems to be outdated, at least in written French.

I know statistics are not authoritative but anyway, Google only finds less than a handful of occurrences of endroits près les uns des autres, two of them being from the eighteenth century, while it finds around twenty times more occurrences of endroits proches les uns des autres, many from modern writings.

A small difference would lead do a different advice though.

Using être situé (to be located) instead of être (to be) mandates the pronoun près:

Tout les endroits que nous allons visiter aujourd’hui sont situés près les uns des autres…

Your last question was already answered by Serge and I fully agree with him.