The SRD does not contradict official rules.

The official (English) SRD rules do not contradict the official rules from the Player's Handbook, Monster Manual or Dungeon Master's Guide, excepting for the names of a few spells changing.

Also remember that the players will likely require your guidance from the official rulebook to create characters, to access certain class features and so-on.

However, the fan-made translations you mentioned might alter the rules as written or someone's interpretation of a rule might translate poorly, which may lead to rules disputes with your players.

There is no rule but there is some guidance you can find

As of yet, there are no rules for nakedness. The section on advantage and disadvantage in the Basic Rules, though, says:

The GM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.

Applying disadvantage could be useful for some things you listed (like Constitution checks for weather and Charisma checks in social situations).

Armor Class change?

Minor cuts and bruises (such that you would get from climbing a tree bare) is usually not accounted for in HP calculations if only for speed of play.

I doubt the added vulnerability of your skin would factor in much. I've always considered the base 10 AC to just be a DC for accuracy in so far as rolling less than 10 means you were off target. Under that interpretation, being unclothed makes no difference.

Furthermore, some methods of calculating armor class are specifically designed for wearing little to no armor (see Unarmored Defense for monks and barbarians). You would be affecting the core of these features by reducing armor class.