It is up to individual countries to sign agreements banning double taxation. If they do not, EU law cannot force them to. And any country you have a connection to can legally require you to complete tax declarations. However, having to declare the same income in 2 EU countries does not necessarily mean you will pay tax twice. And even if you do have to pay tax twice you may be able to reclaim tax in one of the countries.
Under the Spain-Sweden double taxation treaty, you may be exempted from Spanish tax on your Swedish pension or, alternatively, the tax paid in Sweden may be credited against the Spanish tax due on that pension. You will have to consult the tax convention concluded between Spain and Sweden for more information.
Two kinds of issues are worth mentioning here:
Property taxes differ very much between EU countries. It is advisable to seek information on this before buying real estate abroad.
You should also find out about inheritance tax, because laws in this area differ greatly in Europe and there is a very high risk that you might be taxed by two different countries with no remedy for this double taxation.
Ask the tax office of the country in which you intend to buy property.
This would depend on the terms of the bilateral tax agreement between Finland and Italy. Under most bilateral double taxation agreements, your worldwide income, including income from pension investments, would be taxed in your country of residence — which in your case appears to be Italy — but with credit for any tax paid in the country of origin of those investments (Finland in the situation you describe).
There is no EU-wide law on tax issues relating to cross-border pensions. There are only domestic tax laws and tax agreements. In most cases, you only have to pay income tax in the country where you have your primary residence.
If you receive a public-sector pension (if you worked as a civil servant), you will normally have to continue paying taxes on the pension (only) to the country that pays the pension, regardless of your place of residence.
This is only a summary of the most common rule. There could be exceptions to this rule in some international tax conventions. Ask the local tax office(s).
NO — Whichever country is charging the tax, it must charge you the same rate as for domestic dividends. The EU’s equal treatment principle applies to tax on dividends, interest and other securities income, and it applies both in the country where you live and the country where the income is generated.
YES — As you are a resident of Sweden, you are probably indeed liable to Swedish tax on those savings. Furthermore, the French bank may well have correctly withheld tax in accordance with French rules. However, the double tax agreement between France and Sweden will probably mean that France has to refund partially or totally the withholding tax that the bank has applied. Please contact your local tax office in Sweden to check the level of withholding tax that France can apply and the procedure for getting a refund from France of any excess tax applied.
That would depend on the national laws in both countries and on any tax arrangement between them banning double taxation of estates and inheritance. It is possible that you would have to pay tax in both countries but the relevant tax treaty may say UK has to credit inheritance tax paid in France against the tax due in the UK. You will need to contact the French and British tax authorities to establish your rights and obligations.