There are over 40 definitions of socialism. However a common-enough one (SEP) is that:
The bulk of the means of production is under social, democratic control.
And in contrast the corresponding definition of capitalism entails:
The bulk of the means of production is privately owned and controlled.
Under these definition the answer is no, you can't have both socialism and capitalism at the same time in a country. Either the bulk of means of production is in private hands or it's "socially owned". (Actually, that SEP page also allows for a "statism", in which the state owns most of the means of production but the state is not under democratic control. I'm glossing over this distinction between socialism and "statism" here.)
Wikipedia provides a fairly similar definition of socialism, but with a less clear threshold:
Socialism is a range of economic and social systems characterised by social ownership of the means of production and workers' self-management.
Under this definition it's more arguable when something is to be called socialism.
In general, economic publications don't try to put such socialism vs. capitalism labels on countries nowadays. Rather more objective figures like the percentage of state-ownership in the economy and/or the percentage of people employed by the state. For the latter (which is somewhat more objective to calculate as you don't have to try to estimate the value of schools or hospitals) Wikipedia has a list. For example Demark has about 32% of employees in the state sector, whereas it's around 16-19% in the US. For a proper contrast, the page also notes that
In the former Eastern Bloc countries, the public sector in 1989 accounted for between 70% to over 90% of total employment.
Even on such apparently simple employment figures data can be somewhat misleading due to subcontracting
Germany, for example, spends as much as France on health, but has far fewer public employees in the sector. This is explained by the fact that health workers in Germany are generally paid by the administration through private contracts. France, more generally, rarely relies on subcontractors for public services, in contrast to Germany, Denmark and Finland.
So one can be largely paid by the state, but be technically a private employee... See public–private partnership (PPPs) for more on this latter issue. Arguably, PPPs are "less socialist" than outright employment by the state, but some right-wing/libertarian writers nonetheless declare PPPs the "backdoor to socialism"... So I hope you get some idea why answers to your questions depend on definitions.
If you want to move beyond mere ownership and employment by the state, to the state exercise of regulatory power or even dirigisme over the (rest of) the economy, things can get even more complicated. According to some research, the Nordic countries actually exercise less regulatory control than the US so there is more "business freedom" in the Nordic countries according to some indexes, including by OECD, World Bank or (even) the [US-based] Heritage Foundation standards. This despite the higher taxes and social spending in the Nordic countries. Consequently, some argue that a unidimensional scale [never mind a pure dichotomy] of socialism vs capitalism is rather misleading nowadays.