How to best utilise 5k GBP

A lot of this will depend on what you're saving for: retirement, a house deposit, starting a business, etc. - if you're looking at an equity fund (usually, but not always, what people mean when talking index funds) then you'll want enough time to ride out any market dips and wait for the overall trend to assert itself (at least assuming that the last century or so is representative...).

I'm no more than an educated amateur, so don't take this for any kind of actual financial advice - just my thoughts.

1. The broad impression I get here is that you're on a reasonable and sensible track. On the NS&I side, I quite like Premium Bonds for quick access savings - from a pure numbers point of view they aren't the most sensible, but the meaningful impact of £2/month interest is so little that maybe a tiny gamble that doesn't affect the principal is worth it! Vanguard index funds are low fee and do the job well, the key is picking the right one for your needs.

2. I wouldn't. Currency conversions, taxes, and the complexity of dealing with them are likely to wipe out any advantage - if indeed there is an advantage, since you'd be making an active bet against the pound there. A more gentle, and in my opinion more appropriate hedge would be an index fund which contains stocks from a number of countries. You would be exposed to their markets while still keeping the funds in the UK.

3. Generally just let it sit; you might want to tweak the risk exposure as you get closer to the time you want to start withdrawing (e.g. moving a pension fund towards a more bond-heavy mix as you reach retirement), but anything more than that is likely to lose you money either in fees, or in trying and failing to time the market.

4. Look at percentage fees, as well as estimated inflation, and deduct them straight from your expected percentage return. That'll give you an estimated net percentage and you can figure out a real cash expectation from what you put in and how long you intend to leave it there. If you're looking at any fixed amount fees, you can then take them off that cash figure.

5. It's narrowing your field of potential investments, for sure. On the other hand, it's narrowing them in a specific direction that you might think will do better (or worse) than the market at large.

I don't live in the UK so take this advice with some salt:

The usual recommendation is to keep 1-3 month's worth of living expenses as an emergency fund, so 3k may or may not be enough to keep depending on your expenses. I generally direct newbies to the Bogleheads wiki but there may be better resources for the U.K.

1. The only one who can decide the best use of your money is... you. Think on your present and future needs and hold the money accordingly. The time horizon of your plans (emergency fund, saving for a new gadget/house/etc, retirement bucket...) should determine what sort of gains/risk to take.
2. Nobody knows exactly what Brexit will bring, but converting currencies can be expensive and trying to set up bank accounts outside your country of residence can be difficult.
3. An index fund tracking whole markets is a very simple instrument. You don't need to do anything besides purchasing shares (and re-investing the dividends if that's necessary). Some time ago I heard that the U.K. was in a particularly bad place as far as fees went, but if Vanguard is there then their fees should be pretty low.
4. Again, I have no idea what the fee situation in the U.K. is but keeping costs low and being aware of the tax implications of your investments is a necessity.
5. Ethical funds by definition will reduce the diversity of your investments and possibly introduce higher fees (therefore likely to drive down your returns). It is up to you to decide if that's worth it.

Emergency fund is a very good idea. Actually, everyone should have one!

On investing: The potential absolute profits with 2k aren't that big. It may be interesting if you plan on adding additional funds on a regular basis, which greatly increases this potential. On the other hand, it really depends on why you want to invest it. Could you elaborate on that?

It's sensible to keep a cash or highly liquid emergency fund, but pay off any debts you have if they're at interest rates more costly than market rates of return.

In the broader sense, interest on debt is just enriching corporations that are generally behaving more odiously than any good you can do by investing in ethical funds.

Just spend it now before sterling is wrecked at the end of the month. Then at least you'll have some nice stuff to enjoy during the upcoming decade of recession/depression.

If you really must save it, I have some money in various funds at Hargreaves Lansdown who are very helpful if you phone them up and can direct you to some safe/profitable/etc places to stash your cash. Don't just buy shares, that's bonkers. That's why investment managers exist.