1. Brokers such as eToro trade in more than crypto e.g. stocks, currencies. So if I transferred money directly from my bank to a company such as eToro which isn't exclusively crypto, is this likely to be flagged as a banned company from my banks perspective, or is my transaction likely to go through without any queries?
eToro is a established broker like many others around the globe. I see no reason for your bank to consider this as a shady activity since you're basically deciding to take part of your liquidity and use it to invest in the financial produts that are offered by eToro (that can range from stocks, Exchanged Traded Funds and others).
Also, if eToro is then transferring funds back into my bank account, is this likely to be queried? I'm wondering if my bank has the right (I think they probably do?) to ask what I'll be using eToro for and/or what the incoming eToro funds were generated from. If that question was posed to me, I wouldn't want to misinform them by saying 'oh it was trading in stocks' when in reality it came from trading in crypto.
It doesn't make sense for your bank to ask you where the funds came from (because they know that they came from eToro in this case). In my country, for example, you are responsible for presenting your capital gains at the end of your fiscal year (whether you send the gains to your bank account or let them sit at a broker like eToro). So if you decide to cash out your investments to your local bank you should basically do the right thing and report them in your tax reports (assuming that crypto is taxable in your country - I ask you to look for further information on this one).
At the end of the day
it's your money you're talking about. If your bank starts putting restriction on how you should use
YOUR money then I would seriously look for another alternative.
2. To avoid this, am I better to use something like PayPal for my crypto transactions?
I advice strongly against using PayPal (or even Revolut if you're from Europe) for your crypto transactions. The reason is quite simple : They don't let you withdraw your investments (BTC, LTC for example) to a wallet of yours. So you're basically stuck with your investments in a platform that only allows you to trade in their environment which is, by itself, not the wisest decision you could make.
3. If you're comfortable sharing this information, what method do you use for your outgoing and incoming crypto transactions?
In your case, considering you've just arrived to this "world" I would try to create an account at established exchange and do whatever investments you have planned. The main ones are either Coinbase, Kraken or even Binance.
Whenever you can (and considering the amount invested) always try to move your investments out of any CEX (centralized exchange) you may end up using.