How do you increase the space you have in your inventory?

I have a feeling someone will move this over soon but to answer your question the inventory is set to display only a certain number of items at a time and will restrict you if you try to buy an item that might put you over the limit! Gold members do get an extended inventory of 250 items but the limit works the same way. C:

I would recommend clearing out your inventory by moving items you'd like to sell into a shop or storing items you like/might wanna use later into your vault. You can also add wearable items into your wardrobe and put items that you'd like for your pets into their treasure chests!

If you wanna just get rid of or donate them, there is also an option for that! :D Hope this helps!

How many items you can store in your vault?

your vault has unlimited storage.

Unlimited is a relative term. There will be a point beyond which the combination of your browser and this site will cause your Vault to be unworkably slow. The only way to find this point is to mess around with it.

How can you expand the number of pets you can have?

you can buy pet slots in the Millionaire Center or in the Cash Shop. You can participate in site events that give them out, like Quest-A-Thons. Starting your seond year playing here you are granted a Loyalty Box every year and "gimme an extra pet slot" is one of the things you can get from it.