You recalculate skills and saving throws a proficiency bonus is applied to:

• Skill and saving throws based on Str, Dex and Con in which you have proficiency
• Skills and saving throws based on Int, Wis and Cha in which the target beast has proficiency (not common, but perception is the usual case)

You should "extract" the beast's standard proficiency bonus by subtracting the total attack modifier or skill modifier from it's related stat bonus. Example:

The beast has a Strength score of 16, and thus a +3 Str bonus. If it's melee attack modifier is +7, then it's proficiency bonus is 7-3= 4.

Then, If your proficiency bonus is greater than the one extracted, you should use yours to recalculate the skill, saving throws.

As a DM I tell my druid player to keep a record of the adjusted stat block of the 3 most used beasts, and adjust them when his proficiency bonus changes, to speed up the gameplay.

According to Jeremy Crawford, the druid has no proficiency with natural attacks, therefore the attack bonuses should not be recalculated

Skill proficiency and tool proficiency are different things.

You only add your proficiency bonus to a check once. When you take an action that requires a check, if you're proficient in that kind of check, you get to add your proficiency bonus. For example, if you have proficiency in Stealth checks, then you can add your proficiency bonus to any Dexterity (Stealth) rolls you make. If you're proficient with your thieves tools, then you get to add your proficiency bonus when you're trying to pick a lock.

It seems like you're reading the rules on how to apply proficiency bonuses in a way that's more like how D&D 3.5 did it, rather than the way that 5e works. When you pick a lock, you're not making a Stealth check with your tools, you're just making a check with your tools. There is no "skill" involved when you use tools. The only source for a proficiency bonus when using tools is if you are proficient with that tool.

In your example, the Rogue would get a +2 when using the tools. If the Rogue somehow had Stealth proficiency but not thieves tools proficiency, then they wouldn't get any proficiency bonus when using thieves tools.