YES — Most likely. If you stay in a country for longer than 6 months in a year, that country would normally consider you as resident there for tax purposes. Usually, that country will tax the income you earn while working there. You should find out whether you will be considered a tax-resident in your new country and what the applicable rates and tax deductions are. You should also check whether the salary you earn there will be taxed in your home country and what relief, if any, is provided in the double tax agreement between these 2 countries.
Your worldwide income will normally be taxed in the country where you are a tax-resident. To be sure, you should check the bilateral tax agreement between the country where you are a tax-resident and the country from which you derive income.
If you stay in Austria for less than 6 months, you would normally pay tax in Austria on the income you earn from working there, if your employer is resident there. Any income you earn from other sources would be taxed in the UK. To be certain, you should check the UK-Austria bilateral tax convention.
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.
Because you worked in the other country for less than 6 months, that country would normally tax the income you earned there, if your employer is resident there. Income from all sources - including your earnings from the 5 months you worked in the other country - may also be taxed in your home country. However, under the bilateral agreement on double taxation between the two countries, your home country will usually offset the tax paid in the other country against the tax due at home.