Regarding the mechanism for biological activity of tyramine, amphetamine and ephedrine
Question: When these foreign amines have been taken up into the vesicles how do they proceed?...Do they re-enter the synaptic cleft by exocytosis...How are they eventually cleared from the synaptic area?
Here is the mode of action of these agents (indirectly acting sympathomimetics):


The most
important drugs in the indirectly acting sympathomimetic amine
category are tyramine, amphetamine and ephedrine, which are
structurally related to noradrenaline.
Amphetamine enters the nerve terminal via the noradrenaline
transporter (NET) and enters synaptic vesicles via the vesicular
monoamine transporter (VMAT), in exchange for noradrenaline
(NA), which accumulates in the cytosol.
Some of the NA is degraded
by monoamine oxidase (MAO) within the nerve terminal and some escapes,
in exchange for amphetamine via the noradrenaline transporter, to act
on postsynaptic receptors. Amphetamine also reduces NA reuptake via
the transporter, so enhancing the action of the released NA.

Exocytosis is not involved in the release process, so their actions do not require the presence of Ca2+. They are not completely specific in their actions, and act partly by a direct effect on adrenoceptors, partly by inhibiting NET (thereby enhancing the effect of the released noradrenaline) and partly by inhibiting MAO.
These agents are cleared by metabolism via oxidative deamination with monoamine oxidase(MAO) enzyme, however  are poor substrates for MAO and, thus, show a prolonged duration of action.
Amphetamine is an example of a drug metabolized through oxidative deamination:


Question: Why do the noradrenaline re-uptake proteins transport noradrenaline out of the axonal terminal under the influence of these foreign amines?
I am not really sure about the “why” part but instead the “how” part AFAICT, this is more of a displacement mechanism:
These drugs, do block reuptake 1(NET) releasing stored transmitter from noradrenergic nerve endings and they displace vesicular norepinephrine into the cytoplasm. The norepinephrine is then transported out of the nerve terminal by the reverse action of reuptake 1 and activates adrenoceptors.
So essentially the drugs increase the availability of noradrenaline to stimulate adrenergic receptors by:
i. releasing or displacing NE (noradrenaline) from sympathetic nerve endings,
ii. blocking the transport of NE into sympathetic neurons or
iii. blocking the metabolizing enzymes, MAO or COMT.
How this helps
References

Rang and Dale’s Pharmacology 8th Edition. Section 2: Chemical Mediators

Medical Pharmacology at a Glance M. JNeal: Drugs acting on the sympathetic system

Lippincott Illustrated Reviews: Pharmacology 6th Edition: Adrenergic Agonists

Goodman and Gilman Manual of Pharmacology and Therapeutics: Agents Acting at the Neuromuscular Junction and Autonomic Ganglia

Principles of Clinical Pharmacology 2nd Ed Atkinson etal
Basic and Clinical Pharmacology B. Katsung et al.