There are plenty of rules about a Wizard having a second spellbook. But since you're a warlock, none of them apply to you and I won't comment on them here.
Only pact of the tome Warlocks have a grimoire. But you don't copy spells into it, or prepare from it, or do any of the "usual" wizardly things with it.\$^1\$ Specifically:
Warlocks don't prepare spells from a book.
Warlocks simply know some number of spells (and cantrips). When you level up you choose which spells from the Warlock list these are, and that list stays the same until you hit the next level. Since you're Pact of the Tome, you get 3 extra cantrips whenever you're holding that book.
So, to answer the specific questions you posed:
Can I use the book 'as is' by spending the appropriate action to switch between them?
No, you can't use the other book. Your tome was granted you by your patron, and is the only book you "use."
Or would I have to copy the spells from the newly found book into my spell book before using them?
No, you can't copy those spells. That's a wizard thing.\$^1\$
If I can use the book as-is, does that increase the number of spells I can cast per day, or am I still limited based on my character level?
You're not using the book, but if you could any casting spells from it would count against the number of spells you can cast. That number's a feature of your class and level, not of your equipment.
What if there are spells in the new book that are higher level than what I can currently cast? Can I copy them into my book or would I have to wait until I am a high enough level?
You're not using the book, so it doesn't matter what level the spells in it are.\$^1\$
Finally, are there any rules about using the spell book of a warlock from a different master. My patron is a 'great old one'. Are there any special rules that say I can or can not use a spellbook from a warlock who made a pact with a different elder god? What about if they made a pact with an arch-fey or a fiend?
There aren't rules about using spells from a different master. Because no one does. You only have available to you the spells that have been granted you by your master.
\$^1\$ - Unless you have the Book of Ancient Secrets
invocation. (Thank you, @lithas!) In that case you could copy into your grimoire spells you find written from any
class's list, as long as
- they have the "ritual" tag,
- the spell's level is less than half your Warlock level (rounded up),
- you spend the requisite time and materials to copy it.
You can then cast these spells as a ritual whenever you like. As with all rituals, they won't count against your number of spell slots.