The James K answer answers question about inheritance tax but omits the question about welfare. Hence I will focus on that part of your question.
Classical liberals were not opposed to minimal welfare. For example, F. A. Hayek, who was prominent classical liberal in last century stated (Hayek, Consitution of Liberty pp 374) [emphasis mine]:
All modern governments have made provision for the
indigent, unfortunate, and disabled and have concerned themselves with questions of health and the dissemination of knowledge. There is no reason why
the volume of these pure service activities should not increase with the general
growth of wealth. There are common needs that can be satisfied only by collective action and which can be thus provided for without restricting individual liberty. It can hardly be denied that, as we grow richer, that minimum
of sustenance which the community has always provided for those not able
to look after themselves, and which can be provided outside the market, will
gradually rise, or that government may, usefully and without doing any harm,
assist or even lead in such endeavors. There is little reason why the government should not also play some role, or even take the initiative, in such areas
as social insurance and education, or temporarily subsidize certain experimental developments. Our problem here is not so much the aims as the methods of government action. ...
We shall see that some of the aims of the welfare state can be realized
without detriment to individual liberty, though not necessarily by the methods
which seem the most obvious and are therefore most popular; that others can
be similarly achieved to a certain extent, though only at a cost much greater
than people imagine or would be willing to bear, or only slowly and gradually
as wealth increases; and that, finally, there are others—and they are those particularly dear to the hearts of the socialists—that cannot be realized in a society that wants to preserve personal freedom. ...
Here, however, an important distinction has
to be drawn between two conceptions of security: a limited security which can
be achieved for all and which is, therefore, no privilege, and absolute security, which in a free society cannot be achieved for all. The first of these is
security against severe physical privation, the assurance of a given minimum
of sustenance for all; and the second is the assurance of a given standard of
life, which is determined by comparing the standard enjoyed by a person or a
group with that of others. The distinction, then, is that between the security
of an equal minimum income for all and the security of a particular income
that a person is thought to deserve.
Other classical liberals such as Friedman (see Capitalism and Freedom) also advocated for welfare in the form of negative income tax. J.S. Mill also advocated for welfare programs for the poor (see Kurer 1991).
Hence classical liberals were not opposed to welfare state per se. Rather they had completely different conception of welfare state.
Classical liberals did not advocate for welfare state to reduce income, wealth or consumption inequality. Rather they advocated for minimalist welfare state that would ensure nobody falls below certain minimum. Generally, they advocated for something that I would term 'subsistence level welfare'. That is welfare that enables an individual to subsist in society without perishing if an individual is unable to earn enough to subsist on but they would not go very far beyond this. Of course, as in any ideology different classical liberals slightly differ in their position on this issue. For example, Friedman negative income tax would go much further when it comes to redistribution than let's say Hayek's very minimalist view of welfare.
Hence, most classical liberals were not opposed to welfare per se. Rather they advocated for a very simple subsistence level of welfare as opposed to very generous welfare systems, that exist not just for the poorest members of the society, which you can nowadays see in many European states.
PS: There seems to be some widespread confusion about whether Hayek or Friedman count as classical liberals so I want to clear that out.
Friedman and Hayek are sometimes erroneously described as neoliberals due to their influences on neoliberalism and due to poor understanding of taxonomy and research in the field political economy or lack of rigorous. However, majority and virtually all top political economy scholars would consider Friedman and Hayek classical liberals not neoliberals.
This confusion likely arises because both classical liberals and neoliberals have similar thick conception of liberty. However, there is a crucial distinction between them when it comes to the issue of affirming the use social justice in evaluation of society's structure (see the Oxford Handbook of Political Economics pp 117).
As clearly mentioned in the Oxford Handbook of Political Economics (pp 115), which can be considered definitive source on the matter [emphasis mine]:
By "Classical Liberalism" we mean a broad school that includes Adam Smith, David Hume, F.A. Hayek, and libertarians such as Robert Nozick.
The same source also lists M. Friedman as classical liberal elsewhere in the chapter.
In addition, neoliberalism itself is by serious researchers in the area of political economy considered confusing and unclear term. Following Venugopal (2015):
[Neoliberalism] is now widely acknowledged in the literature as a controversial, incoherent, and crisis-ridden term, even by many of its most influential deployers,
As a result the word neoliberalism is often thrown around incorrectly by people who do not have rigorous background/education in political economics.
In fact Venugopal (2015) even explicitly mentions:
There are indeed significant points of linkage and overlap in the neoliberalisms before and after 1980, including for example the link between the Chicago
School and Chile’s Chicago Boys, and the inspiration that Hayek is reputed to
have provided for Margaret Thatcher. But leaving aside the tendency of scholars
to read too much into the influence of scholarly ideas on policy and politicians,
these connections are weak and often based on exaggerated projections of the
present into the past. The history of neoliberalism as a moniker does not always
correspond to the genealogy of a radical free-market project, and this association,
which seems self-evident in the ‘neoliberal present’, has come about through a
series of haphazard and contingent processes.
Beyond conceptual proliferation and incoherence, there is an important third
terminological feature of neoliberalism that more clearly distinguishes it from
the multitude of other stressed and stretched concepts that dot the social
sciences: it dares not speak its own name. While there are many who give out
and are given the title of neoliberal, there are none who will embrace this
moniker of power and call themselves as such. There is no contemporary body
of knowledge that calls itself neoliberalism, no self-described neoliberal
theorists that elaborate it, nor policy-makers or practitioners that implement
it. There are no primers or advanced textbooks on the subject matter, no
pedagogues, courses or students of neoliberalism, no policies or election
manifestoes that promise to implement it (although there are many that
promise to dismantle it). Pedantic as it may seem, this is a point that warrants
repetition if only because there is a considerable body of critical literature that
deploys neoliberalism under the mistaken assumption that, in doing so, it is
being transported into the front-lines of hand-to-hand combat with freemarket economics.
Indeed the term neoliberalism is rarely used in actual mainstream scholarship on political economics. However, when it is used it is not primarily associated with Hayek and Friedman, but rather with other more modern Chicago School economics. Usually the distinction is drawn between thick liberal thinkers who rejected the idea of social justice and thinkers who embraced it. Moreover, the term neoliberalism itself is often inconsistently applied, so one can occasionally see confused mentions of Hayek and Friedman as neoliberal or they might be part of some heterodox and not widely accepted (by most Political Economy scholars) definitions of neoliberalism, but rigorous and generally accepted scholarship (e.g. Oxford Handbook of Political Economy) denotes these thinkers as classical liberal.