I can give you the Italian answer - first of all, normally we don't use tomato paste to make sauce, but rather to add a tomato "kick" to recipes. Tomato paste is simply tomato puree that has been cooked down to a high degree of concentration.
A basic tomato sauce is made by
- making a soffritto with onion, carrot and celery (plus other flavors)
- adding tomato puree, or "pelati", more rarely fresh tomato
- cooking the sauce down until the taste and thickness is what you want
If you want to use paste instead of puree, the third step has to be omitted or greatly reduced in duration. Keep in mind that tomato paste has its own taste, and that taste will remain in the final sauce.
The best way I've found to get full flavor out of the mint is to infuse your liquor with it. I do this for mint juleps. Simply take a handful of mint, bruise it (you can just crush it in your hand or stick it in a bag and whack it a couple of times with a wooden spoon), and place it in your liquor of choice (white rum in this case) and leave it for a day or two. You'll need to experiment with the right amount for your drink, but I loosely pack a mason jar about half full to do bourbon.
If you want the actual leaves in your drink, then you need to make sure you muddle the mint rather than just putting it in. What you want from the mint is the oils in it, these are released better when you bruise the leaves. So just chopping and throwing it in won't work well. You actually need to crush them.
It's also important to use the lime rind in addition to the juice. And again, muddle it. The reasoning is the same. There are oils in the rind that are released during muddling that have a decidedly different flavor than just the juice.
To finish out, pick your sweetener. I like to use simple syrup, but you can also use sugar (or anything sweet, if you want to change the flavor, try a flavored syrup). And then top it up with something non-alcoholic. I particularly like sprite, but also use lemonade or sometimes just soda water. Be careful to balance the sweetener based on whatever you're adding; Sprite obviously needs less sugar added than soda water.
Best Answer
Mint likes oil. And it likes water. And it likes alcohol. Like most complex flavors, mint is complicated.
The greener, vegetal notes are going to be from compounds like chlorophyll, and will be alcohol and very weakly water soluble. The astringent, sharper notes are going to from compounds like menthol, which are oil soluble.
In general, the faster flavors, the ones that hit fast and fade faster, are water soluble, while the ones that linger are oil soluble. Mint flavor is sold both as an extract, in a base of water or alcohol, and as an essential oil, in a base of... oil.
As for cooking, I would skip the tomatoes entirely, and do a mint and parsley pesto with walnuts and olive oil. Why? I like mint.