How long will homemade salad dressing stay good in the fridge?

There is no rule of thumb encompassing all salad dressings.

One "rule of thumb" which sometimes gets used is to look at the shelf life of the most perishable component. Frequently it works; sometimes, it is dangerously misleading. A mixture of the things you listed can have a longer or shorter shelf life of that of the most perishable component. An example of lengthened shelf life is mayo; it stays good for longer time than a cracked fresh egg, because the yolk gets pasteurized and the pH is lower. But there can be examples of the other thing happening. The classic is the homemade garlic oil: you can keep pure garlic and pure oil for months in the pantry, but once you combine them, you get a botulism risk.

I don't usually keep salad dressings in the fridge, but if I did, I would look at why the most problematic component has the shelf life it has, and decide whether mixing it will change the condition. For example, if I had a mixture of oil, vinegar and pure emulsifier: Oil keeps for months because it has no carbs, so nothing for bacteria to eat. Vinegar can have a few carbs, but not a lot, and it also has a very low pH, so bacteria die in it. The combination still won't have enough carbs for bacteria, and will still have a low pH. So the mixture will keep for very long time, just like pure vinegar or oil.

On the other hand, imagine mixing vinegar with honey and water. Honey doesn't go bad by itself, because the carbs are too concentrated for bacteria. The vinegar and water will dilute them, so this mechanism of bacteria prevention vanishes. The pH of the mixture will probably rise a lot too, because the vinegar gets diluted, so no protection on that front either. Thus, this mixture is apt to go bad much earlier than pure vinegar or pure honey.

This method requires that you make a new decision for each new dressing you make, and that you acquire enough knowledge to be able to make such decisions. If you feel this is too much effort or too risky, you can either start adding conservants to your homemade dressings, or just start preparing a fresh dressing for each batch of salad.

The quickest and dirtiest rule of thumb is thus: "The more acidic it is, the longer it will last." Acid is very unfriendly to bacteria. Ketchup, vinagrette, fruit syrups, all fine. Yes, fruit syrup, even if it's just sugar and acid, will last quite a while (think fruit jelly, and fruit preserves.) Homemade pepper sauce. Anything with a bunch of alcohol in it.

Mayonnaise is scary for about a dozen reasons: it's not just the oil, or the eggs, or the sugar...It's all of them. Anything with dairy in it...That always comes with its own set of bacteria to jumpstart the process of decay. Anything with too many raw vegetables in it (pesto, for example).

If you are using raw garlic, I'd recommend keeping it no more than a few days given the risk of botulism. This risk can be mitigated (although not completely avoided) by using dried garlic or garlic powder, heating the dressing (which is undesirable if you are using a fruity oil), or making the dressing very acidic (which is likely also undesirable).

The only other particularly perishable ingredient is the mayonnaise, especially if it is homemade. I'd say that, garlic aside, you should use any homemade mayonnaise-based dressing within six days or so.